Question

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

choose the correct sketch of a + b below.

a.

b.

c.

d.

Explanation:

Step1: Recall Vector Addition Rule

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

Step2: Analyze Each Option

• Option A: The vectors \(\mathbf{a}\) and \(\mathbf{b}\) are placed with \(\mathbf{a}\) starting at a point, and \(\mathbf{b}\) starting at the head of \(\mathbf{a}\), but the resultant \(\mathbf{a}+\mathbf{b}\) is drawn as a dashed line from the tail of \(\mathbf{a}\) to the head of \(\mathbf{b}\)? Wait, no, looking at the diagram, in the triangle law, if we have \(\mathbf{a}\) (vertical) and \(\mathbf{b}\) (slanted), placing the tail of \(\mathbf{b}\) at the head of \(\mathbf{a}\) would make the resultant from tail of \(\mathbf{a}\) to head of \(\mathbf{b}\). Wait, no, let's check the correct representation. The correct way for vector addition (triangle law) is: \(\mathbf{a}+\mathbf{b}\) is obtained by \(\mathbf{a}\) (from origin to point \(A\)) and \(\mathbf{b}\) (from \(A\) to point \(B\)), so \(\mathbf{a}+\mathbf{b}\) is from origin to \(B\). Alternatively, parallelogram: \(\mathbf{a}\) and \(\mathbf{b}\) with same initial point, diagonal is \(\mathbf{a}+\mathbf{b}\).

Looking at the options:

• Option D: The vectors \(\mathbf{a}\) (vertical) and \(\mathbf{b}\) (slanted) are placed with the same initial point, and the diagonal of the parallelogram (or the triangle) is drawn as \(\mathbf{a}+\mathbf{b}\), which matches the vector addition (either triangle or parallelogram law). Let's verify: If we take \(\mathbf{a}\) as the vertical vector and \(\mathbf{b}\) as the slanted vector, placing them tail - to - tail, the resultant \(\mathbf{a}+\mathbf{b}\) should be the diagonal from the common tail to the opposite corner of the parallelogram formed by \(\mathbf{a}\) and \(\mathbf{b}\). In Option D, the vectors \(\mathbf{a}\) and \(\mathbf{b}\) are tail - to - tail, and the resultant \(\mathbf{a}+\mathbf{b}\) is the diagonal, which is correct.

• Option A: The resultant is drawn in a way that does not follow the vector addition law (the dashed line and the direction seem off).
• Option B: The resultant is drawn from the head of \(\mathbf{a}\) to the head of \(\mathbf{b}\), which is not correct for vector addition.
• Option C: The vectors are arranged in a way that the resultant would be \(\mathbf{b}-\mathbf{a}\) or some other combination, not \(\mathbf{a}+\mathbf{b}\).

Answer:

D. The sketch in Option D (where vectors \(\mathbf{a}\) and \(\mathbf{b}\) are tail - to - tail and the resultant \(\mathbf{a}+\mathbf{b}\) is the diagonal from the common tail to the opposite corner)