Question

for the vectors u and w with angle θ between them sketch the resultant. |u|=25, |w|=12, θ=28° choose the correct sketch of the resultant below. a. image of vector diagram b. image of vector diagram c. image of vector diagram

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 if we place the tail of one vector at the head of the other, the resultant is the vector from the tail of the first to the head of the second. The parallelogram law states that the resultant is the diagonal of the parallelogram formed by the two vectors.

Step2: Analyze the Magnitudes and Angle

We know \( |\mathbf{u}| = 25 \), \( |\mathbf{w}| = 12 \), and \( \theta = 28^\circ \). The angle between \( \mathbf{u} \) and \( \mathbf{w} \) is \( 28^\circ \), so when sketching the resultant, we need to check the direction and the relative lengths.

Step3: Evaluate the Sketches

- **Option A**: The vectors \( \mathbf{w} \) and \( \mathbf{u} \) are placed tail - to - tail, and the resultant is drawn using the triangle law (from tail of \( \mathbf{w} \) to head of \( \mathbf{u} \) when \( \mathbf{w} \) is placed at tail of \( \mathbf{u} \)? Wait, no, in option A, we have \( \mathbf{w} \) with a small magnitude (since \( |\mathbf{w}| = 12 \) and \( |\mathbf{u}| = 25 \), \( \mathbf{w} \) should be shorter than \( \mathbf{u} \)) and \( \mathbf{u} \) longer. The angle between them is \( 28^\circ \), and the resultant is drawn correctly as the vector from the tail of \( \mathbf{w} \) to the head of \( \mathbf{u} \) when \( \mathbf{w} \) is placed at the tail of \( \mathbf{u} \)? Wait, no, let's re - examine. Wait, the length of \( \mathbf{u} \) is 25, \( \mathbf{w} \) is 12, so \( \mathbf{u} \) should be longer than \( \mathbf{w} \). In option A, \( \mathbf{u} \) is longer than \( \mathbf{w} \), the angle between them is \( 28^\circ \), and the resultant is the red vector. But wait, let's check the other options.
- **Option B**: The vectors are placed tail - to - tail, and the resultant is the diagonal of the parallelogram. But the length of \( \mathbf{w} \) and \( \mathbf{u} \) seem to be misrepresented ( \( \mathbf{w} \) looks almost as long as \( \mathbf{u} \), but \( |\mathbf{w}| = 12 \) and \( |\mathbf{u}| = 25 \), so \( \mathbf{w} \) should be shorter).
- **Option C**: The direction of the vectors and the resultant seem incorrect. The resultant should be in a direction that combines the two vectors.

Wait, actually, when using the triangle law, if we have vectors \( \mathbf{w} \) and \( \mathbf{u} \) with \( \mathbf{w} \) having a smaller magnitude (12) than \( \mathbf{u} \) (25), and an angle of \( 28^\circ \) between them, the correct sketch should show \( \mathbf{w} \) shorter than \( \mathbf{u} \) and the resultant vector. But looking at the options, in option A, \( \mathbf{w} \) is shorter than \( \mathbf{u} \), the angle between them is \( 28^\circ \), and the resultant is drawn as the vector from the tail of \( \mathbf{w} \) to the head of \( \mathbf{u} \) (when we place \( \mathbf{w} \) at the tail of \( \mathbf{u} \), but actually, the triangle law is: place the tail of \( \mathbf{w} \) at the head of \( \mathbf{u} \), then the resultant is from tail of \( \mathbf{u} \) to head of \( \mathbf{w} \)? No, I think I made a mistake. The correct way is: for vector addition \( \mathbf{u}+\mathbf{w} \), we can place the tail of \( \mathbf{w} \) at the head of \( \mathbf{u} \), then the resultant \( \mathbf{u}+\mathbf{w} \) is the vector from the tail of \( \mathbf{u} \) to the head of \( \mathbf{w} \). In option A, we have \( \mathbf{w} \) with a smaller length (since \( |\mathbf{w}| = 12 \), \( |\mathbf{u}| = 25 \)), the angle between \( \mathbf{u} \) and \( \mathbf{w} \) is \( 28^\circ \), and the resultant is drawn correctly. Wait, but the original problem says "sketch the resultant". The key is the relative lengths and the angle. Since \( |\mathbf{u}| = 25 \) is longer than \( |\mathbf{w}| = 12 \), and the angle between them is \( 28^\circ \), the correct sketch should have \( \mathbf{u} \) longer than \( \mathbf{w} \) and the resultant vector. Among the options, option A has \( \mathbf{u} \) longer than \( \mathbf{w} \), the angle between them is \( 28^\circ \), and the resultant is drawn as the vector from the tail of \( \mathbf{w} \) to the head of \( \mathbf{u} \) (using the triangle law). Wait, maybe I was confused earlier. Let's re - check:

The magnitude of \( \mathbf{u} \) is 25, \( \mathbf{w} \) is 12, so \( \mathbf{u} \) is longer. In option A, the black vector \( \mathbf{u} \) is longer than \( \mathbf{w} \), the angle between them is \( 28^\circ \), and the red resultant is drawn from the tail of \( \mathbf{w} \) to the head of \( \mathbf{u} \) (which is the triangle law for \( \mathbf{w}+\mathbf{u} \)). In option B, \( \mathbf{w} \) and \( \mathbf{u} \) seem to be of similar length, which is incorrect. In option C, the direction is wrong. So the correct sketch is option A? Wait, no, the user has option C selected, but maybe I made a mistake. Wait, let's think again. The angle between \( \mathbf{u} \) and \( \mathbf{w} \) is \( \theta = 28^\circ \). When we sketch the resultant, using the parallelogram method, we place the vectors tail - to - tail, and the resultant is the diagonal. But in the triangle method, we place them head - to - tail. Wait, the length of \( \mathbf{u} \) is 25, \( \mathbf{w} \) is 12, so \( \mathbf{u} \) is longer. In option A, \( \mathbf{w} \) is the shorter vector, \( \mathbf{u} \) is longer, angle between them is \( 28^\circ \), and the resultant is drawn. In option B, \( \mathbf{w} \) is almost as long as \( \mathbf{u} \), which is wrong. In option C, the resultant is in the wrong direction. So the correct answer should be option A? Wait, maybe the initial selection by the user was wrong. Let's re - evaluate.

Wait, the problem says "For the vectors u and w with angle θ between them sketch the resultant". The angle between them is \( 28^\circ \), \( |\mathbf{u}| = 25 \), \( |\mathbf{w}| = 12 \). So \( \mathbf{u} \) is longer than \( \mathbf{w} \). In option A, the black vector \( \mathbf{u} \) is longer than \( \mathbf{w} \), the angle between them is \( 28^\circ \), and the red resultant is drawn from the tail of \( \mathbf{w} \) to the head of \( \mathbf{u} \) (triangle law). In option B, the vectors are tail - to - tail, but \( \mathbf{w} \) is too long. In option C, the resultant is in the opposite direction. So the correct sketch is option A. But maybe I messed up the vector labels. Wait, in option A, the black vector is \( \mathbf{u} \), the other black is \( \mathbf{w} \), red is resultant. Yes, so the correct answer is option A.

Answer:

A. The sketch with \( \mathbf{w} \) (shorter vector) and \( \mathbf{u} \) (longer vector) with an angle of \( 28^\circ \) between them, and the resultant vector drawn from the tail of \( \mathbf{w} \) to the head of \( \mathbf{u} \) (triangle law of vector addition).