Questions: Find the component form of v given its magnitude and the angle it makes with the positive x-axis. Sketch v. Magnitude Angle v=7 theta=45°

Find the component form of v given its magnitude and the angle it makes with the positive x-axis. Sketch v. Magnitude Angle v=7 theta=45°
Transcript text: Find the component form of $\mathbf{v}$ given its magnitude and the angle it makes with the positive $x$-axis. Sketch $\mathbf{v}$. Magnitude Angle \[ \|\mathbf{v}\|=7 \quad \theta=45^{\circ} \]
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Solution

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Solution Steps

Step 1: Find the x-component of the vector

The x-component of vector v is given by ||v||cos θ = 7 * cos(45°) = 7 * (√2/2) = (7√2)/2.

Step 2: Find the y-component of the vector

The y-component of vector v is given by ||v||sin θ = 7 * sin(45°) = 7 * (√2/2) = (7√2)/2.

Step 3: Write the component form and select the correct graph

The component form of vector v is v = <(7√2)/2, (7√2)/2>. Since both components are positive and equal, the vector lies along the line y = x in the first quadrant, making a 45° angle with the positive x-axis. The second graph from the top is the correct representation.

Final Answer:

v = <(7√2)/2, (7√2)/2>, second graph from the top.

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