Free Vectors and Equilibrium MCQs with Answers

1. Two forces of 3 N and 4 N act at right angles to each other. The magnitude of their resultant is

• A. 1 N
• B. 5 N
• C. 7 N
• D. 12 N

Explanation: For perpendicular vectors the resultant is the square root of (3 squared plus 4 squared), which is the square root of 25, giving 5 N. Adding the magnitudes to get 7 N would only be correct if the forces acted in the same direction, and 1 N is the result if they were exactly opposed. Every answer between 1 N and 7 N is possible depending on the angle.

2. Two forces of equal magnitude F act at an angle of 120 degrees to each other. Their resultant has magnitude

• A. 2F
• B. F times the square root of 3
• C. F
• D. zero

Explanation: Using the parallelogram rule, the resultant is the square root of (F squared plus F squared plus 2F squared cos 120), and since cos 120 is minus one half the last term cancels one of the others, leaving the square root of F squared, which is F. Equal vectors at 120 degrees are the arrangement in which three such vectors would sum to zero. At 0 degrees the resultant would be 2F and at 180 degrees it would be zero.

3. The minimum number of unequal coplanar forces whose vector sum can be zero is

• A. one
• B. two
• C. three
• D. four

Explanation: Two unequal forces can never cancel, since cancelling requires equal magnitudes in opposite directions. Three forces can close a triangle head to tail even when all three magnitudes differ, so three is the minimum. This is why the triangle of forces is the standard construction for a body in equilibrium under three forces.

4. A vector of magnitude 20 N acts at 30 degrees above the horizontal. Its horizontal component is

• A. 10 N
• B. 20 N
• C. 17.3 N
• D. 11.5 N

Explanation: The horizontal component is 20 cos 30, and cos 30 is 0.866, so the component is 17.3 N. The vertical component is 20 sin 30, which is 10 N, and that is the value offered as the distractor. Note that the components are always smaller than the vector itself, which is a quick sanity check.

5. If a vector has equal rectangular components, it makes an angle with the x axis of

• A. 30 degrees
• B. 45 degrees
• C. 60 degrees
• D. 90 degrees

Explanation: The angle is given by the inverse tangent of the ratio of the y component to the x component, and when the two are equal that ratio is 1, whose inverse tangent is 45 degrees. Each component then has magnitude equal to the vector divided by the square root of 2. At 90 degrees the x component would be zero rather than equal to the y component.

6. The scalar product of two vectors A and B is defined as

• A. AB sin theta, and the result is a vector
• B. AB cos theta, and the result is a scalar
• C. AB tan theta, and the result is a scalar
• D. AB, independent of the angle between them

Explanation: The dot product multiplies the magnitudes by the cosine of the angle between the vectors and gives a scalar, which is why work, the dot product of force and displacement, is a scalar. The sine form belongs to the vector product, whose result is a vector. The cosine factor is what makes the dot product zero for perpendicular vectors.

7. The scalar product of two perpendicular vectors is

• A. maximum
• B. equal to their magnitudes multiplied
• C. zero
• D. negative

Explanation: Cos 90 is zero, so the dot product vanishes whenever the vectors are at right angles, and this is the standard test for perpendicularity. It is also why no work is done by a force acting at right angles to the displacement, such as the centripetal force on a body in circular motion. The dot product is maximum when the vectors are parallel and negative when the angle exceeds 90 degrees.

8. The vector product of two parallel vectors is

• A. zero
• B. maximum and equal to AB
• C. equal to AB cos theta
• D. a scalar

Explanation: The cross product has magnitude AB sin theta, and sin 0 is zero, so parallel vectors give a null vector. It is maximum for perpendicular vectors, which is the opposite of the dot product and the easiest way to keep the two straight. The result of a cross product is always a vector, never a scalar.

9. The direction of the vector product A cross B is given by

• A. the direction of A
• B. the direction of B
• C. the bisector of the angle between A and B
• D. the right hand rule, perpendicular to the plane containing A and B

Explanation: Curling the fingers of the right hand from A to B points the thumb along A cross B, which is perpendicular to both vectors. Because the sense reverses when the order is swapped, B cross A equals minus A cross B, so the cross product is not commutative. Torque and magnetic force are both defined this way.

10. The first condition of equilibrium requires that

• A. the vector sum of all forces acting on the body is zero
• B. the sum of all torques about any point is zero
• C. the body is at rest
• D. no force acts on the body at all

Explanation: With zero net force there is no acceleration of the centre of mass, which is translational equilibrium. The second condition deals with rotation and requires zero net torque, and a body needs both to be in complete equilibrium. Note that equilibrium allows motion at constant velocity, so being at rest is not required.