Question

Vector v has a magnitude of 23 and a direction angle of 270°. Write this vector as a linear combination of the standard unit vectors i and j.
(1 point)
Responses

v = 248.5i + 105.5j
v = 248.5 i + 105.5 j

bold v equals 248.5 bold i minus 105.5 bold j
Image with alt text: bold v equals 248.5 bold i minus 105.5 bold j

bold v equals negative 23 bold i
Image with alt text: bold v equals negative 23 bold i

bold v equals negative 23 bold i

Answer 1

To express the vector \( \mathbf{v} \) in terms of the standard unit vectors \( \mathbf{i} \) and \( \mathbf{j} \), we can use the magnitude and the direction angle. The magnitude of the vector \( \|\mathbf{v}\| \) is given to be 23, and the direction angle is 270°, which corresponds to the negative y-axis.

The components of the vector can be calculated as follows:
- The x-component is given by \( v_x = \|\mathbf{v}\| \cdot \cos(\theta) \)
- The y-component is given by \( v_y = \|\mathbf{v}\| \cdot \sin(\theta) \)

Substituting the values we have:
- \( \theta = 270^\circ \)
- \( \|\mathbf{v}\| = 23 \)

1. Calculate the x-component:
\[
v_x = 23 \cdot \cos(270^\circ) = 23 \cdot 0 = 0
\]

2. Calculate the y-component:
\[
v_y = 23 \cdot \sin(270^\circ) = 23 \cdot (-1) = -23
\]

Therefore, the vector \( \mathbf{v} \) can be written as:
\[
\mathbf{v} = 0 \mathbf{i} - 23 \mathbf{j}
\]

This means the vector \( \mathbf{v} \) can be simplified to:
\[
\mathbf{v} = -23 \mathbf{j}
\]

So the correct representation of vector \( \mathbf{v} \) among the given responses is:
**bold v equals negative 23 bold j**