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Scalars and Vectors
Understand scalar and vector quantities, position vector, displacement vector, unit vector, vector multiplication, dot product, cross product and vector components.
Scalar Quantities
A scalar quantity is a physical quantity that has magnitude only and no direction. Scalars are completely described by a number and a unit, such as 5 kg, 20 s or 300 K.
- Magnitude only means the size or numerical value is enough.
- No direction is attached to a scalar quantity.
- A scalar can be positive, negative or zero depending on the physical meaning.
- A positive quantity is not automatically a scalar. For example, a force of 10 N east is positive in magnitude but still a vector because direction is present.
- Common mistake: treating distance and displacement as the same. Distance is scalar, displacement is vector.
Quick Scalar Test
If changing direction changes the meaning of the answer, it is not scalar. If only numerical size with unit is enough, it is scalar.
| Scalar Quantity | Unit Example | Why It Is Scalar |
|---|---|---|
| Mass | kg | Only amount of matter is required. |
| Time | s | No direction is associated with duration. |
| Temperature | K or °C | Only hotness or coldness value is needed. |
| Energy | J | Energy has magnitude but no direction. |
| Work | J | Work is a scalar product of force and displacement. |
| Distance | m | It measures total path length only. |
| Speed | m s-1 | It gives how fast, not where. |
| Electric potential | V | Potential is energy per unit charge, a scalar. |
Vector Quantities
A vector quantity is a physical quantity that has both magnitude and direction. It is represented by an arrow: the length of the arrow shows magnitude, and the arrowhead shows direction.
- The starting point of the arrow is called the tail.
- The ending pointed part is called the head.
- Vector notation may be written as bold A, arrow A, or component form A = Ax î + Ay ĵ.
Displacement Velocity Acceleration Force Momentum Electric field Magnetic field
Position Vector
The position vector of a point is the vector drawn from the origin to that point. If a point P has coordinates (x, y), its 2D position vector is written using unit vectors along the x and y axes.
In 3D, the z-coordinate is added along k̂. The position vector changes if the origin changes.
Displacement Vector
The displacement vector is the vector drawn from the initial point to the final point. It depends only on the starting and ending positions, not on the path followed.
The position vector tells where a point is from the origin. The displacement vector tells the change in position from one point to another.
Equality of Vectors
Two vectors are equal when they have the same magnitude and the same direction. Their location does not matter, so equal vectors may be drawn at different places in a diagram.
- Same length means same magnitude.
- Parallel and same sense means same direction.
- If length differs or direction differs, vectors are unequal.
Unit Vector
Definition
A unit vector has magnitude 1 and shows direction only. It is used to describe direction without changing the size of a vector.
Formula
â = A / |A|
Here |A| is the magnitude of vector A.
Standard Unit Vectors
î is along x-axis, ĵ is along y-axis and k̂ is along z-axis. Each has magnitude 1.
Example: If A = 3î + 4ĵ, then |A| = 5, so the unit vector along A is â = (3/5)î + (4/5)ĵ.
Negative Vector
The negative of a vector has the same magnitude but the opposite direction. If A is east, then -A is west with the same size.
Multiplication of Vector by Scalar
When a vector is multiplied by a scalar, its magnitude changes. If the scalar is positive, direction remains the same. If the scalar is negative, direction becomes opposite.
- 2A has twice the magnitude of A in the same direction.
- 3A has three times the magnitude of A in the same direction.
- -2A has twice the magnitude of A in the opposite direction.
Resolution of Vector into Components
Resolution means splitting a vector into mutually perpendicular components. For a vector A making angle θ with the x-axis, the horizontal component lies along x-axis and the vertical component lies along y-axis.
Rectangular components help solve projectile motion, inclined plane, relative velocity and force problems by treating x and y directions separately.
Derivation of R2 = A2 + B2 + 2AB cos θ
This derivation is very important for CBSE and school-level vector addition. It gives the magnitude of the resultant of two vectors A and B making angle θ with each other.
R2 = (A + B cos θ)2 + (B sin θ)2
R2 = A2 + 2AB cos θ + B2 cos2 θ + B2 sin2 θ
R2 = A2 + B2 + 2AB cos θ
CBSE Exam Tip: For θ = 90°, cos θ = 0, so R2 = A2 + B2. For θ = 0°, R = A + B. For θ = 180°, R = |A - B|.
Dot Product
The dot product of two vectors gives a scalar result. It measures how much one vector acts along another vector.
- Dot product is commutative: A · B = B · A.
- It is maximum when θ = 0°.
- It is zero when θ = 90°.
- It is negative when θ > 90°.
Unit Vector Products
î · î = 1 ĵ · ĵ = 1 k̂ · k̂ = 1 î · ĵ = 0
Applications: work done, power, projection and magnetic flux.
Dot Product and Cross Product: Full Worked Example
Let A = î + 2ĵ - 3k̂ and B = 3î - 3ĵ + 4k̂. We will find A · B, A × B and the angle θ between the two vectors.
Angle Using Dot Product
Given: A = î + 2ĵ - 3k̂, B = 3î - 3ĵ + 4k̂
Dot product: A · B = (1)(3) + (2)(-3) + (-3)(4)
Calculation: A · B = 3 - 6 - 12 = -15
Magnitudes: |A| = √(12 + 22 + (-3)2) = √14, |B| = √(32 + (-3)2 + 42) = √34
Formula: A · B = |A||B| cos θ
Result: cos θ = -15 / √476 = -15 / (2√119)
Final Answer: θ ≈ 133.4°
Exam Tip: Negative dot product means the angle is obtuse.
Cross Product by Determinant
Expansion: A × B = î[(2)(4) - (-3)(-3)] - ĵ[(1)(4) - (-3)(3)] + k̂[(1)(-3) - (2)(3)]
Calculation: A × B = î(8 - 9) - ĵ(4 + 9) + k̂(-3 - 6)
Final Cross Product: A × B = -î - 13ĵ - 9k̂
Magnitude: |A × B| = √(1 + 169 + 81) = √251
Angle Check: sin θ = |A × B| / |A||B| = √251 / √476. This gives acute reference angle ≈ 46.6°. Since dot product is negative, actual angle is 180° - 46.6° = 133.4°.
Cross Product
The cross product of two vectors gives a vector result. Its magnitude depends on the sine of the angle between the vectors, and its direction is found by the right-hand thumb rule.
- It is anti-commutative: A × B = -(B × A).
- It is zero when vectors are parallel.
- It is maximum when vectors are perpendicular.
- Applications: torque, angular momentum and magnetic force.
î × ĵ = k̂ ĵ × k̂ = î k̂ × î = ĵ ĵ × î = -k̂
Important Examples
Example 1: Identify scalar and vector quantities from speed, force, distance and velocity.
Given: Speed, force, distance, velocity.
Formula/Idea: Scalars have magnitude only; vectors have magnitude and direction.
Calculation: Speed and distance do not need direction. Force and velocity need direction.
Final Answer: Scalars: speed, distance. Vectors: force, velocity.
Exam Tip: Do not confuse speed with velocity.
Example 2: Find position vector of point P(3, 4).
Given: x = 3, y = 4.
Formula: r = x î + y ĵ.
Calculation: r = 3î + 4ĵ.
Final Answer: Position vector is 3î + 4ĵ.
Exam Tip: Position vector is always measured from the origin.
Example 3: Find displacement from A(1, 2) to B(5, 7).
Given: r1 = 1î + 2ĵ, r2 = 5î + 7ĵ.
Formula: Δr = r2 - r1.
Calculation: Δr = (5 - 1)î + (7 - 2)ĵ = 4î + 5ĵ.
Final Answer: 4î + 5ĵ.
Exam Tip: Always subtract initial position from final position.
Example 4: Find unit vector along A = 6î + 8ĵ.
Given: A = 6î + 8ĵ.
Formula: â = A / |A|.
Calculation: |A| = √(62 + 82) = 10. So â = (6/10)î + (8/10)ĵ.
Final Answer: â = 0.6î + 0.8ĵ.
Exam Tip: A unit vector must have magnitude 1.
Example 5: Resolve a 10 N vector at 30° with x-axis.
Given: A = 10 N, θ = 30°.
Formula: Ax = A cos θ, Ay = A sin θ.
Calculation: Ax = 10 cos 30° = 5√3 N; Ay = 10 sin 30° = 5 N.
Final Answer: Components are 5√3 N and 5 N.
Exam Tip: Angle with x-axis uses cos for x-component.
Example 6: Find dot product of A = 2î + 3ĵ and B = 4î + ĵ.
Given: A = 2î + 3ĵ, B = 4î + ĵ.
Formula: A · B = AxBx + AyBy.
Calculation: A · B = 2 × 4 + 3 × 1 = 11.
Final Answer: 11.
Exam Tip: Dot product result is scalar.
Example 7: Find cross product of î and ĵ.
Given: î and ĵ are perpendicular unit vectors.
Formula: î × ĵ = k̂.
Calculation: Using cyclic order î, ĵ, k̂.
Final Answer: k̂.
Exam Tip: Reversing the order changes the sign.
Example 8: Find angle between equal vectors when A · B = A B / 2.
Given: A · B = AB/2.
Formula: A · B = AB cos θ.
Calculation: AB cos θ = AB/2, so cos θ = 1/2.
Final Answer: θ = 60°.
Exam Tip: Cancel magnitudes only if both vectors are non-zero.
Numericals
CBSE: A vector of magnitude 20 makes 60° with x-axis. Find components.
Given: A = 20, θ = 60°.
Formula: Ax = A cos θ, Ay = A sin θ.
Calculation: Ax = 20 × 1/2 = 10; Ay = 20 × √3/2 = 10√3.
Final Answer: Ax = 10, Ay = 10√3.
Exam Tip: Memorize sin 60° and cos 60°.
NEET: Find resultant of two perpendicular vectors 6 N and 8 N.
Given: A = 6 N, B = 8 N, θ = 90°.
Formula: R = √(A2 + B2).
Calculation: R = √(36 + 64) = 10 N.
Final Answer: 10 N.
Exam Tip: Perpendicular vectors form a right triangle.
JEE Main: If A = 3î + 4ĵ and B = 4î - 3ĵ, calculate A · B.
Given: A = 3î + 4ĵ, B = 4î - 3ĵ.
Formula: A · B = AxBx + AyBy.
Calculation: A · B = 3 × 4 + 4 × (-3) = 12 - 12 = 0.
Final Answer: 0, so vectors are perpendicular.
Exam Tip: Zero dot product means 90° angle for non-zero vectors.
JEE Advanced: Find |A × B| if |A| = 5, |B| = 12 and θ = 30°.
Given: A = 5, B = 12, θ = 30°.
Formula: |A × B| = AB sin θ.
Calculation: |A × B| = 5 × 12 × 1/2 = 30.
Final Answer: 30.
Exam Tip: Cross product magnitude uses sin θ.
IB Physics: A student walks 3 m east and 4 m north. Find displacement.
Given: Components are 3 m east and 4 m north.
Formula: R = √(x2 + y2), tan θ = y/x.
Calculation: R = 5 m, tan θ = 4/3.
Final Answer: 5 m at tan-1(4/3) north of east.
Exam Tip: Distance is 7 m but displacement is 5 m.
IGCSE: A car travels with velocity 15 m s-1 east. Is it scalar or vector?
Given: Magnitude 15 m s-1 and direction east.
Formula/Idea: Velocity has magnitude and direction.
Calculation: Direction is specified, so it is a vector.
Final Answer: Vector quantity.
Exam Tip: Speed is scalar; velocity is vector.
A-Level: Find unit vector along 2î - ĵ + 2k̂.
Given: A = 2î - ĵ + 2k̂.
Formula: â = A / |A|.
Calculation: |A| = √(4 + 1 + 4) = 3.
Final Answer: â = (2/3)î - (1/3)ĵ + (2/3)k̂.
Exam Tip: Include signs of components carefully.
PYQ Section
The following are solved PYQ-style and authentic-year examples. Exact years are mentioned only where commonly known; otherwise they are marked as exam-style questions.
NEET 2018: Which physical quantity is a scalar?
Given: Options commonly compare force, velocity, displacement and work.
Formula/Idea: Work is the dot product of force and displacement.
Calculation: Dot product gives scalar result.
Final Answer: Work.
Exam Tip: Work may be positive, negative or zero, but it is scalar.
JEE Main 2020: If two non-zero vectors have zero dot product, find the angle between them.
Given: A · B = 0.
Formula: A · B = AB cos θ.
Calculation: Since A and B are non-zero, cos θ = 0.
Final Answer: θ = 90°.
Exam Tip: Zero dot product indicates perpendicular vectors.
JEE Advanced 2019: Direction of A × B is decided by which rule?
Given: Cross product direction.
Formula/Idea: Use right-hand thumb rule.
Calculation: Curl fingers from A to B; thumb gives A × B direction.
Final Answer: Right-hand thumb rule.
Exam Tip: B × A points in the opposite direction.
CBSE Exam-style Question: Define unit vector and write its formula.
Given: Vector A.
Formula: â = A / |A|.
Calculation: Divide the vector by its magnitude.
Final Answer: A unit vector has magnitude 1 and direction of A.
Exam Tip: Unit vector is dimensionless if it only represents direction.
IB Physics Exam-style Question: Explain why displacement is vector but distance is scalar.
Given: Distance and displacement.
Formula/Idea: Vector needs direction; scalar does not.
Calculation: Distance is total path length. Displacement is shortest directed change in position.
Final Answer: Displacement is vector; distance is scalar.
Exam Tip: Use a path diagram if asked for explanation.
IGCSE Exam-style Question: State one scalar and one vector quantity.
Given: Examples required.
Formula/Idea: Scalar has magnitude only; vector has magnitude and direction.
Calculation: Mass is scalar. Force is vector.
Final Answer: Scalar: mass. Vector: force.
Exam Tip: Avoid writing velocity as scalar.
A-Level Exam-style Question: What is the vector product of parallel vectors?
Given: θ = 0° or 180°.
Formula: |A × B| = AB sin θ.
Calculation: sin 0° = 0 and sin 180° = 0.
Final Answer: Zero vector.
Exam Tip: Cross product is maximum for perpendicular vectors.
Large PYQ-Based Practice Bank
These are high-frequency, PYQ-based vector practice questions for revision. No fake years are added. Use them for NEET, JEE Main, JEE Advanced, IB and ICSE/IGCSE style preparation.
NEET Practice Bank: 50 Vector Questions
- Identify scalar quantities from work, velocity, force and displacement.
- Identify vector quantities from speed, distance, acceleration and mass.
- Find resultant of two perpendicular vectors 3 and 4.
- Find resultant of two vectors 5 and 5 at 60°.
- Find resultant of two equal vectors at 120°.
- Find angle when resultant of two equal vectors equals either vector.
- Find x and y components of a 10 N force at 30°.
- Find x and y components of a velocity 20 m s-1 at 60°.
- Find unit vector along 3î + 4ĵ.
- Find unit vector along 2î - 2ĵ + k̂.
- Calculate dot product of 2î + 3ĵ and î - 2ĵ.
- Find angle if A · B = 0 for non-zero A and B.
- Find angle if A · B = AB.
- Find angle if A · B = -AB.
- Calculate work done by force 10 N through displacement 5 m at 60°.
- Find projection of A on B when A · B and |B| are given.
- Calculate cross product magnitude for A = 4, B = 5 and θ = 90°.
- Find cross product magnitude for θ = 0°.
- State direction of î × ĵ.
- State direction of ĵ × î.
- Find A + B if A = 2î + 3ĵ and B = î - ĵ.
- Find A - B if A = 5î - 2ĵ and B = 3î + ĵ.
- Find magnitude of 6î + 8ĵ.
- Find magnitude of î + 2ĵ + 2k̂.
- Find direction angle of vector √3î + ĵ.
- Find direction angle of vector î + √3ĵ.
- Compare distance and displacement for a closed path.
- Decide whether electric potential is scalar or vector.
- Decide whether electric field is scalar or vector.
- Find resultant of 10 N east and 10 N north.
- Find resultant of 8 N east and 6 N west.
- Find vector opposite to 4î - 3ĵ.
- Find 2A if A = î - 2ĵ + 3k̂.
- Find -3A if A = 2î + ĵ.
- Find A · A in terms of |A|.
- Find A × A for any vector A.
- Find whether two vectors are perpendicular using dot product.
- Find whether two vectors are parallel using cross product.
- Find components of weight on an inclined plane.
- Find horizontal and vertical components of projectile velocity.
- Use parallelogram law to find resultant magnitude.
- Use triangle law to add two displacement vectors.
- Find displacement after 3 m east and 4 m south.
- Find angle between 2î + ĵ and î + 2ĵ.
- Find dot product when angle is 120°.
- Find cross product magnitude when angle is 30°.
- Find torque magnitude using rF sin θ.
- Find if momentum is scalar or vector.
- Find if temperature gradient direction makes a quantity vector.
- Choose correct statement about scalar multiplication of a vector.
JEE Main / IIT Practice Bank: 50 Vector Questions
- Find resultant magnitude of A and B using R2 = A2 + B2 + 2AB cos θ.
- Derive condition for resultant to be minimum.
- Derive condition for resultant to be maximum.
- Find angle between A = 2î + 3ĵ and B = 4î - ĵ.
- Find projection of A = 3î + 4ĵ on x-axis.
- Find projection of A on B using A · B / |B|.
- Find vector component of A along B.
- Find area of parallelogram using |A × B|.
- Find area of triangle using 1/2 |A × B|.
- Find A × B for A = î + ĵ and B = î - ĵ.
- Find A × B for A = 2î - k̂ and B = ĵ + k̂.
- Find scalar triple product meaning for coplanarity.
- Check if three vectors are coplanar using determinant.
- Find vector perpendicular to two given vectors.
- Find unit vector perpendicular to A and B.
- Find λ if A = î + λĵ and B = 2î - 3ĵ are perpendicular.
- Find λ if A = 2î + λĵ and B = 4î + 6ĵ are parallel.
- Find resultant of three vectors in component form.
- Find equilibrant of a given resultant vector.
- Find angle bisector direction of two equal vectors.
- Find velocity of rain relative to a moving man using vector subtraction.
- Find boat velocity relative to river current.
- Find shortest crossing condition in river-boat vector problem.
- Find drift in river crossing using component method.
- Find resultant acceleration from tangential and centripetal components.
- Find relative velocity of two cars moving at right angles.
- Find relative velocity of two particles from position vectors.
- Differentiate position vector to get velocity vector.
- Differentiate velocity vector to get acceleration vector.
- Find speed from velocity vector components.
- Find displacement vector from r(t) at two times.
- Find direction of motion from velocity vector.
- Find dot product in terms of components in 3D.
- Find cross product in determinant form in 3D.
- Find angle between diagonals of a parallelogram.
- Find condition for two vectors to have equal magnitude.
- Find unit vector along angle bisector of two unit vectors.
- Find magnitude of A + B and A - B and relate them.
- Use (A + B) · (A - B) to compare magnitudes.
- Find A · B from |A + B| and |A - B|.
- Find angle if |A + B| = |A - B|.
- Find resultant of vectors inclined at 30°, 60° and 90°.
- Find component of gravity along inclined plane.
- Find component of normal reaction in vector form.
- Find work done by variable direction force using dot product idea.
- Find torque vector direction for r and F.
- Find angular momentum direction using r × p.
- Find magnetic force direction using v × B.
- Find if vector product is distributive over addition.
- Compare scalar and vector products for same two vectors.
JEE Advanced / IIT Advanced Practice Bank: 35 Higher-Level Questions
- Find all possible angles between A and B if |A + B| = √3 |A - B|.
- Given |A| = 2, |B| = 3 and |A × B| = 3, find possible A · B values.
- If A · B = A · C and A × B = A × C, discuss relation between B and C.
- Find unit vector perpendicular to 2î + ĵ - k̂ and î - 2ĵ + 3k̂.
- Find area of triangle with vertices represented by three position vectors.
- Use vector method to prove diagonals of a parallelogram bisect each other.
- Find shortest distance from point to line using vector product idea.
- Find λ such that three vectors become coplanar.
- Evaluate (A + B) × (A - B) and interpret direction.
- Evaluate (A + B) · (A - B) and interpret when it is zero.
- Find A and B angle if |A + B| = |A| = |B|.
- Find A and B angle if |A - B| = |A| = |B|.
- Find resultant of three equal vectors symmetrically placed at 120°.
- Find resultant of three non-coplanar unit vectors î, ĵ, k̂.
- Find maximum and minimum of |A + B + C| under given magnitudes.
- Find projection of a vector on a plane using normal vector.
- Find vector rejection of A from B.
- Find angle between A × B and B × A.
- Find angle between A × B and A.
- Find angle between A × B and B.
- Find |A × B|2 + (A · B)2 in terms of A and B.
- Prove Lagrange identity for two vectors using components.
- Find vector equation of median in a triangle using position vectors.
- Find centroid position vector of a triangle.
- Find condition for four points to form a parallelogram using position vectors.
- Find torque due to multiple forces using vector sum.
- Find angular momentum vector for given r and p.
- Find magnetic force vector on charge for given v and B.
- Find work done when force and displacement vectors are given in components.
- Find power using F · v.
- Find flux using E · A.
- Find sign of dot product from an obtuse angle diagram.
- Find sign of cross product from orientation of two vectors.
- Find direction cosines of a vector and verify their square sum.
- Find vector with given magnitude and direction ratios.
IB Physics Vector Practice Questions
- Distinguish between distance and displacement with an example.
- Resolve a 12 N force into horizontal and vertical components.
- Find resultant displacement after two perpendicular journeys.
- Use vector addition to solve aircraft wind velocity problem.
- Explain why velocity is vector and speed is scalar.
- Calculate work done using Fd cos θ.
- Find component of weight down an inclined plane.
- Use scale drawing idea to add two vectors.
- Find relative velocity of two moving objects.
- Explain vector nature of acceleration in circular motion.
ICSE / IGCSE Physics Vector Practice Questions
- State two scalar quantities and two vector quantities.
- Explain why force is a vector quantity.
- Explain why mass is a scalar quantity.
- Find resultant of two forces acting in the same direction.
- Find resultant of two forces acting in opposite directions.
- Find resultant of two perpendicular displacements.
- Draw vector diagram for 3 m east and 4 m north.
- Differentiate between speed and velocity.
- Calculate displacement from a simple map diagram.
- Identify whether acceleration due to gravity is scalar or vector.
Assertion Reason
A: Speed is scalar. R: Speed has magnitude only.
Answer: Both A and R are true, and R correctly explains A.
Explanation: Speed does not require direction, so it is scalar.
A: Unit vector has magnitude one. R: Unit vector represents direction.
Answer: Both A and R are true, and R explains the purpose of a unit vector.
Explanation: Dividing a vector by its magnitude leaves only direction.
A: Ax = A sin θ when θ is measured from x-axis. R: Horizontal component is adjacent to θ.
Answer: A is false, R is true.
Explanation: When θ is from x-axis, Ax = A cos θ and Ay = A sin θ.
A: Dot product of perpendicular vectors is zero. R: cos 90° = 0.
Answer: Both A and R are true, and R correctly explains A.
Explanation: A · B = AB cos θ, so for θ = 90°, the result is zero.
A: Cross product is anti-commutative. R: A × B and B × A have opposite directions.
Answer: Both A and R are true, and R correctly explains A.
Explanation: A × B = -(B × A).
Case Study Questions
Case Study 1: Displacement vector in 2D motion
Passage: A particle moves from A(2, 1) to B(8, 7) in the xy-plane.
Questions: 1. Write r1. 2. Write r2. 3. Find Δr. 4. Find magnitude of displacement.
Answers: r1 = 2î + ĵ; r2 = 8î + 7ĵ; Δr = 6î + 6ĵ; |Δr| = 6√2.
Explanation: Use Δr = r2 - r1 and magnitude formula.
Case Study 2: Force components on inclined plane
Passage: A force F acts at angle θ to the horizontal on a block.
Questions: 1. Horizontal component? 2. Vertical component? 3. What if θ = 0°? 4. Why components are useful?
Answers: Fx = F cos θ; Fy = F sin θ; at 0°, Fx = F and Fy = 0; components allow separate x-y analysis.
Explanation: Components are projections of the force along axes.
Case Study 3: Work done using dot product
Passage: A force of 20 N moves a body through 5 m at 60° to the force.
Questions: 1. Which product is used? 2. Formula? 3. Calculate work. 4. Is work scalar?
Answers: Dot product; W = Fs cos θ; W = 20 × 5 × 1/2 = 50 J; yes, work is scalar.
Explanation: Only the component of force along displacement does work.
Case Study 4: Torque using cross product
Passage: Torque is produced when force acts at a distance from the axis of rotation.
Questions: 1. Vector formula? 2. Magnitude? 3. When is torque maximum? 4. Direction rule?
Answers: τ = r × F; |τ| = rF sin θ; maximum at 90°; direction by right-hand thumb rule.
Explanation: Torque is a cross product, so perpendicular force is most effective.
Case Study 5: Vector addition in navigation
Passage: A boat moves 4 km east and then 3 km north.
Questions: 1. Write resultant vector. 2. Find magnitude. 3. Find direction expression. 4. Is total distance same as displacement?
Answers: R = 4î + 3ĵ km; |R| = 5 km; tan θ = 3/4; distance is 7 km, displacement is 5 km.
Explanation: Vector addition uses components, while distance follows actual path.
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