Test on Vectors

Question 1: The magnitude of the 3D vector **b** = (1, -2, 5) is exactly 6 units.

A. Yes

B. No

Explanation: To calculate the magnitude of a 3D vector **v** = (v₁, v₂, v₃), the formula is |**v**| = √(v₁² + v₂² + v₃²). For the vector **b** = (1, -2, 5), its magnitude is |**b**| = √(1² + (-2)² + 5²) = √(1 + 4 + 25) = √30. Since √30 is approximately 5.477, it is not exactly 6 units.

Question 2: According to the provided study materials, which of the following quantities can be fully described by a single numerical value representing its size or amount, without an associated direction?

A. Displacement

B. Velocity

C. Temperature

D. Acceleration

Explanation: The study materials define a scalar quantity as one that can be fully described by a single numerical value, representing its size or amount, and notes that these quantities do not have an associated direction. Temperature is explicitly listed as an example of a scalar quantity. Displacement, velocity, and acceleration are all listed as examples of vector quantities, which possess both magnitude and direction.

Question 3: Is the cross product anti-commutative, which means that **a** × **b** is equal to - (**b** × **a**)?

A. Yes

B. No

Explanation: The study materials state under the properties of the cross product that **a** × **b** = - (**b** × **a**), explicitly labeling this property as anti-commutative.

Question 4: Which statement(s) accurately describe the effect of multiplying a vector by a scalar, based on the provided study materials?

A. If the scalar k is positive, the direction of the resulting vector k​v is the same as the original vector v.

B. If the scalar k is negative, the direction of the resulting vector k​v is opposite to the original vector v.

C. Multiplying a vector by any scalar always changes both its magnitude and its direction.

D. If the scalar k is zero, the resulting vector k​v is the zero vector.

Explanation: According to the study materials: * 'If k > 0, the direction of k​v is the same as v.' This confirms option 0. * 'If k < 0, the direction of k​v is opposite to v.' This confirms option 1. * 'If k = 0, then k​v is the zero vector (0, 0, 0).' This confirms option 3. * Option 2 is incorrect because the material states that scalar multiplication 'changes the vector's magnitude and potentially its direction.' The word 'potentially' implies it does not always change direction (e.g., if k=1, direction is unchanged) and if k=1 or k=-1, the magnitude may not change in the sense of scaling. More critically, if k=0, the direction is not well-defined, and the vector becomes the zero vector, not necessarily just a changed magnitude and direction in the sense of a non-zero vector.

Question 5: The scalar triple product of vectors **a**, **b**, and **c** is defined as the scalar quantity obtained from **b** · (**a** × **c**).

A. Yes

B. No

Explanation: The study materials define the scalar triple product as **a** · (**b** × **c**), not **b** · (**a** × **c**).