6. vector \\( \\vec{a} \\) has a magnitude of 29 units and points in the positive y direction. when vector…
6. 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} \\).
Answer
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.