Question

for the vectors u and w with angle θ between them sketch the resultant.

u

= 10,

w

= 22, θ = 36°

choose the correct sketch of the resultant below.
a.
image of vector diagram a

b.
image of vector diagram b

c.
image of vector diagram c

Explanation:

Step1: Recall Vector Addition

To find the resultant of two vectors \(\mathbf{u}\) and \(\mathbf{w}\) with an angle \(\theta\) between them, we use the parallelogram law or the triangle law of vector addition. The triangle law states that we place the tail of one vector at the head of the other, and the resultant is from the tail of the first to the head of the second. The parallelogram law involves placing the vectors tail - to - tail and the resultant is the diagonal of the parallelogram formed.

Given \(|\mathbf{u}| = 10\), \(|\mathbf{w}|=22\), and \(\theta = 36^{\circ}\) between them. When using the parallelogram method, we place the two vectors \(\mathbf{u}\) and \(\mathbf{w}\) tail - to - tail. The angle between them is \(36^{\circ}\), and the resultant vector (the diagonal of the parallelogram) should be drawn from the common tail to the opposite corner of the parallelogram.

Looking at the options:

• Option A: The vectors \(\mathbf{u}\) and \(\mathbf{w}\) are placed tail - to - tail, and the resultant (red vector) is the diagonal of the parallelogram formed by \(\mathbf{u}\) and \(\mathbf{w}\). The angle between \(\mathbf{u}\) and \(\mathbf{w}\) is also consistent with \(36^{\circ}\) (a non - right, acute angle).
• Option B: This seems to be using a triangle law but the orientation and the angle representation do not match the given angle between \(\mathbf{u}\) and \(\mathbf{w}\) as the angle between \(\mathbf{u}\) and \(\mathbf{w}\) in the triangle is not \(36^{\circ}\) (it looks more like a right - angled or obtuse - angled situation).
• Option C: The vector addition here is not following the correct parallelogram or triangle law for the given vectors \(\mathbf{u}\) and \(\mathbf{w}\) with angle \(36^{\circ}\) between them. The orientation of the vectors and the resultant is incorrect.

Answer:

A. The sketch with vectors \(u\) and \(w\) tail - to - tail, forming a parallelogram with the resultant (red vector) as the diagonal.