Question

given the vectors a and b, sketch the vector a + b

choose the correct sketch of a + b below.
a.
sketch of option a
b.
sketch of option b
c.
sketch of option c
d.
sketch of option d

Explanation:

Step1: Recall Vector Addition Rule

To add two vectors \(\mathbf{a}\) and \(\mathbf{b}\), we can use the triangle law or the parallelogram law of vector addition. The triangle law states that we place the tail of vector \(\mathbf{b}\) at the head of vector \(\mathbf{a}\), and the resultant \(\mathbf{a}+\mathbf{b}\) is the vector from the tail of \(\mathbf{a}\) to the head of \(\mathbf{b}\). The parallelogram law states that if we place the tails of \(\mathbf{a}\) and \(\mathbf{b}\) together, the resultant is the diagonal of the parallelogram formed by \(\mathbf{a}\) and \(\mathbf{b}\).

Step2: Analyze Each Option

• Option A: The vectors \(\mathbf{a}\) and \(\mathbf{b}\) have their tails together (parallelogram law setup), and the diagonal from the common tail to the opposite corner is labeled \(\mathbf{a}+\mathbf{b}\). This matches the parallelogram law of vector addition.
• Option B: The resultant vector is not correctly formed as the direction and the law of addition (triangle or parallelogram) are not followed here.
• Option C: The vectors are arranged in a way that does not follow either the triangle or parallelogram law for addition.
• Option D: The labeling and the arrangement of vectors do not conform to the standard vector addition laws.

Answer:

A. The sketch where vectors \(\mathbf{a}\) and \(\mathbf{b}\) have their tails together, and the diagonal from the common tail to the opposite corner is labeled \(\mathbf{a}+\mathbf{b}\)