Free Vectors and Equilibrium MCQs with Answers

• A. -2
• B. 2
• C. 4
• D. Both A & B

Explanation: The magnitude is |A| = √(2² + 1² + a²) = √(5 + a²). Setting this equal to 3 gives 5 + a² = 9, so a² = 4 and a = ±2.

• A. Zero
• B. Two
• C. Three
• D. Infinite

Explanation: In three-dimensional space, a vector can be resolved into components along the x-, y- and z-axes.

• A. Zero
• B. Two
• C. Three
• D. Infinite

Explanation: A vector in a plane can be resolved into two perpendicular components, usually along the x- and y-axes.

• A. 1st
• B. 2nd
• C. Both 1st and 3rd
• D. 4th

Explanation: The x and y components have the same sign in the first quadrant, where both are positive, and in the third quadrant, where both are…

• A. First
• B. Second
• C. Third
• D. Both 1st and 3rd

Explanation: Tangent is positive when sine and cosine have the same sign. This occurs in the first quadrant, where both are positive, and the third…

• A. 2i^ - 2j^ + k^
• B. (2i^ - 2j^ +k^)/9
• C. (2i^ - 2j^ +k^)/3
• D. (2i^ - 2j^ +k^)/5

Explanation: The magnitude of A = 2i - 2j + k is sqrt((2)² + (-2)² + (1)²) = sqrt9 = 3.

• A. Vector
• B. Position Vector
• C. Vector in the direction at angle θ with an x-axis
• D. The unit vector in the direction at angle θ with an x-axis

Explanation: The magnitude of cosθ i + sinθ j is sqrt(cos²θ + sin²θ) = sqrt1 = 1, so it is a unit vector.

• A. 1
• B. 2
• C. 3
• D. 4

Explanation: In three-dimensional Cartesian space, a vector may have three mutually perpendicular rectangular components along the x-, y- and z-axes.

• A. Polar coordinate system
• B. Cartesian coordinate system
• C. Cylindrical coordinate system
• D. Spherical coordinate system

Explanation: The rectangular coordinate system uses mutually perpendicular axes and is called the Cartesian coordinate system, after René Descartes.

• A. Displacement
• B. Speed
• C. Velocity
• D. Acceleration

Explanation: Speed is a scalar because it has magnitude only, whereas displacement, velocity and acceleration also require direction and are vectors.

• A. 1
• B. i^
• C. j^
• D. k^

Explanation: Using the unit-vector cross-product rule, j x k = i, and therefore i·(j x k) = i·i = 1.

• A. A.B = A^ B^
• B. |A| = |B|
• C. |A^| = |B^|
• D. AB^ = BA^

Explanation: If A and B are equal vectors, then their magnitudes and unit vectors are equal, and A B-hat = B A-hat also remains equal.

• A. A.B/A
• B. A.B/B
• C. A.cosθ
• D. Both a and b are correct

Explanation: The scalar projection of B on A is the component of B along the direction of A, given by A · B divided by the magnitude of A, written here…

• A. A. B = Ax Bx + Ay By + Az Bz
• B. A. B = Ax By + Ay Bz + Az By
• C. A. B = Ay Bz + Az By + Az Bx
• D. A. B = Ax Bz + Ay By + Az Bx

Explanation: The dot product multiplies corresponding components and then adds them: A · B = AxBx + AyBy + AzBz.

• A. 0
• B. 1
• C. -1
• D. i^

Explanation: Using the unit-vector cross-product rule, j × k = i. Hence i · (j × k) = i · i = 1, because the dot product of a unit vector with itself…

• A. 30°
• B. 45°
• C. 60°
• D. 180°

Explanation: For vectors of magnitudes A and B, the cross-product magnitude is AB sin θ and the dot-product magnitude is AB cos θ.

• A. West
• B. North
• C. Down
• D. Up

Explanation: Take east as +i, north as +j, and up as +k; south is therefore -j. East x south = i x (-j) = -(i x j) = -k, which points down.

• A. Head to tail rule
• B. Right-hand rule
• C. Left-hand rule
• D. Triangular rule

Explanation: The direction of a vector product is perpendicular to both original vectors and is found by curling the fingers of the right hand from the…

• A. Zero
• B. One
• C. i^
• D. k^

Explanation: The unit vectors follow the right-hand rule: i × j = k. The vectors are perpendicular, so the cross product has magnitude 1 × 1 × sin90° =…

• A. Same as
• B. Perpendicular to
• C. Opposite to
• D. Inclined to

Explanation: The negative of a vector has a direction that is opposite to that of the original vector.