Question

1. vector \\( \vec{a} \\) has a magnitude of 29 units and points in the positive y direction. when vector \\( \vec{b} \\) is added to \\( \vec{a} \\), the resultant vector \\( \vec{a} + \vec{b} \\) points in the negative y direction with a magnitude of 14 units. find the magnitude and direction of \\( \vec{b} \\).

Explanation:

Step1: Define vectors in y-axis

Let positive y-direction be $+\hat{y}$.
$\vec{A} = 29\hat{y}$, $\vec{A}+\vec{B} = -14\hat{y}$

Step2: Solve for $\vec{B}$

Rearrange $\vec{B} = (\vec{A}+\vec{B}) - \vec{A}$
$\vec{B} = -14\hat{y} - 29\hat{y} = -43\hat{y}$

Step3: Find magnitude of $\vec{B}$

Magnitude is absolute value of the component:
$|\vec{B}| = |-43| = 43$

Step4: Identify direction of $\vec{B}$

The vector is $-43\hat{y}$, so it points in the negative y-direction.

Answer:

The magnitude of $\vec{B}$ is 43 units, and it points in the negative y-direction.