Two vectors have magnitudes of 10 and 15.The angle between them when they are drawn with their tails at the same point is 65 $\degree$.The component of the longer vector along the line of the shorter is:
A)0
B)4.2
C)6.3
D)9.1
E)14

Question

Quizplus · Accepted Answer

First, we need to determine the angle between the two vectors in radians:

$65^\circ = \frac{65\pi}{180} 	ext{ radians} \approx 1.1345	ext{ radians}$

Next, we can use the dot product formula to find the component of the longer vector along the line of the shorter:

$\vec{A} \cdot \vec{B} = |\vec{A}||\vec{B}|\cos	heta$

where $\vec{A}$ is the longer vector, $\vec{B}$ is the shorter vector, $|\vec{A}| = 15$, $|\vec{B}| = 10$, and $	heta$ is the angle between them.

$\vec{A} \cdot \vec{B} = (15)(10)\cos(1.1345) \approx 63$

This dot product represents the projection of $\vec{A}$ onto the line of $\vec{B}$. To find the component of $\vec{A}$ along the line of $\vec{B}$, we need to divide by the magnitude of $\vec{B}$:

$	ext{component of }\vec{A}	ext{ along line of }\vec{B} = \frac{\vec{A} \cdot \vec{B}}{|\vec{B}|} \approx 6.3$

Therefore, the answer is (C) 6.3.

Two Vectors Have Magnitudes of 10 and 15 °\degree° The Component of the Longer Vector Along the Line of Them

Two Vectors Have Magnitudes of 10 and 15 $\degree$ The Component of the Longer Vector Along the Line of Them