Test on Scalar multiplication and collinearity

Question 1: If a vector 'a' is multiplied by a scalar 'k', the magnitude of the resulting vector 'k·a' is always equal to 'k' times the magnitude of the original vector 'a'.

A. Yes

B. No

Explanation: The study materials state that the magnitude of the new vector 'k·a' is the absolute value of the scalar 'k' multiplied by the magnitude of the original vector 'a', expressed as |k·a| = |k| · |a|. This means if 'k' is a negative scalar, the magnitude would be positive (|k| · |a|), not negative (k · |a|).

Question 2: When a three-dimensional vector `a = (a₁, a₂, a₃)` undergoes scalar multiplication by `k`, the resulting vector is represented as `(k·a₁, k·a₂, k·a₃)`.

A. Yes

B. No

Explanation: According to the study materials, for a three-dimensional vector `a = (a₁, a₂, a₃)`, the scalar multiple `k·a` is indeed given by `(k·a₁, k·a₂, k·a₃)`. Scalar multiplication in component form involves multiplying each component of the vector by the scalar.

Question 3: If a vector 'a' has a magnitude of 5 units, and it is multiplied by a scalar 'k = -3', what will be the magnitude of the resulting vector 'k·a'?

A. 5 units

B. 15 units

C. -15 units

D. 0 units

Explanation: The study materials state that the magnitude of the new vector 'k·a' is the absolute value of the scalar 'k' multiplied by the magnitude of the original vector 'a', expressed as |k·a| = |k| · |a|. Given |a| = 5 and k = -3, the magnitude of the resulting vector is |-3| · |5| = 3 · 5 = 15 units.

Question 4: According to the provided study materials, when a non-zero vector is multiplied by a scalar of zero, what is the effect on its direction?

A. The resulting vector retains the original direction.

B. The resulting vector points in the opposite direction.

C. The resulting vector has no specific direction.

D. The resulting vector's direction becomes perpendicular to the original.

Explanation: The study materials state that 'If k = 0 (zero scalar): The result is the zero vector 0·a = (0, 0). The zero vector has no magnitude and no specific direction'. Therefore, multiplying a vector by a zero scalar results in a vector with no specific direction.

Question 5: Two non-zero vectors `a` and `b` are collinear only if they point in the same direction.

A. Yes

B. No

Explanation: Two non-zero vectors `a` and `b` are collinear if they lie on the same line or on parallel lines, which means they can point in the same direction or in exactly opposite directions. The study materials explicitly state, 'If `k` is negative, `a` and `b` point in opposite directions' while still being collinear (since `b = k·a` where `k` is a negative scalar).