Question

given the vectors a and b, sketch the vector a - b.

choose the correct sketch of a - b below.

a.
sketch of vector a, vector b, and vector a - b

b.
sketch of vector a, vector -b, and vector a - b

c.
sketch of vector a, vector -b, and vector a - b

d.
sketch of vector a, vector -b, and vector a - b

Explanation:

Step1: Recall Vector Subtraction Rule

Vector subtraction \( \mathbf{a} - \mathbf{b} \) is equivalent to \( \mathbf{a} + (-\mathbf{b}) \), where \( -\mathbf{b} \) is the vector with the same magnitude as \( \mathbf{b} \) but opposite direction.

Step2: Analyze Vector Addition for \( \mathbf{a} - \mathbf{b} \)

To find \( \mathbf{a} - \mathbf{b} \), we place the tail of \( -\mathbf{b} \) at the tip of \( \mathbf{a} \) (or vice - versa, depending on the method) and the resultant vector \( \mathbf{a}-\mathbf{b} \) is from the tail of \( \mathbf{a} \) to the tip of \( -\mathbf{b} \) when we use the triangle law of vector addition for \( \mathbf{a}+(-\mathbf{b}) \).
Looking at the options:

• Option A: The vector \( \mathbf{a}-\mathbf{b} \) is drawn as the diagonal of the parallelogram formed by \( \mathbf{a} \) and \( \mathbf{b} \), but this is for \( \mathbf{a}+\mathbf{b} \) (parallelogram law), not \( \mathbf{a}-\mathbf{b} \).
• Option B: We have vector \( \mathbf{a} \) and vector \( -\mathbf{b} \) (opposite of \( \mathbf{b} \)). When we apply the triangle law for \( \mathbf{a}+(-\mathbf{b}) \), the resultant vector \( \mathbf{a}-\mathbf{b} \) is correctly drawn from the tail of \( \mathbf{a} \) to the tip of \( -\mathbf{b} \) (after placing the tail of \( -\mathbf{b} \) at the tip of \( \mathbf{a} \)).
• Option C: The configuration of vectors does not follow the correct rule for \( \mathbf{a}-\mathbf{b}=\mathbf{a}+(-\mathbf{b}) \).
• Option D: The arrangement of vectors \( \mathbf{a} \) and \( -\mathbf{b} \) and the resultant vector is incorrect.

Answer:

B. The sketch where vector \( \mathbf{a} \) and \( -\mathbf{b} \) are used to form \( \mathbf{a}-\mathbf{b} \) correctly (as per the triangle law of vector addition for \( \mathbf{a}+(-\mathbf{b}) \))