Question

the vector x has magnitude 7 and direction angle 84°. the vector y has magnitude 4. find the magnitude and direction angle θ of the sum x + y. round your answers to the nearest thousandth. |x + y| = θ = °

Explanation:

Step1: Recall vector - addition formula for magnitude

The magnitude of the sum of two vectors $\vec{x}$ and $\vec{y}$ is given by $|\vec{x}+\vec{y}|=\sqrt{|\vec{x}|^{2}+|\vec{y}|^{2}+2|\vec{x}||\vec{y}|\cos(\theta_{x}-\theta_{y})}$, where $|\vec{x}|$ and $|\vec{y}|$ are the magnitudes of $\vec{x}$ and $\vec{y}$ respectively, and $\theta_{x}$ and $\theta_{y}$ are their direction - angles. Here, $|\vec{x}| = 7$, $|\vec{y}| = 4$, $\theta_{x}=84^{\circ}$, and $\theta_{y}=0^{\circ}$.

Step2: Calculate the cosine of the angle difference

First, find $\cos(\theta_{x}-\theta_{y})=\cos(84^{\circ}-0^{\circ})=\cos(84^{\circ})\approx0.1045$.

Step3: Substitute values into the magnitude formula

$|\vec{x}+\vec{y}|=\sqrt{7^{2}+4^{2}+2\times7\times4\times0.1045}=\sqrt{49 + 16+5.852}=\sqrt{70.852}\approx8.4174$.

Step4: Recall the formula for the direction angle of the resultant vector

The formula for the direction angle $\theta$ of the resultant vector $\vec{R}=\vec{x}+\vec{y}$ is $\tan\theta=\frac{|\vec{x}|\sin\theta_{x}+|\vec{y}|\sin\theta_{y}}{|\vec{x}|\cos\theta_{x}+|\vec{y}|\cos\theta_{y}}$. Substitute $|\vec{x}| = 7$, $\theta_{x}=84^{\circ}$, $|\vec{y}| = 4$, and $\theta_{y}=0^{\circ}$.
$\sin\theta_{x}=\sin(84^{\circ})\approx0.9945$, $\cos\theta_{x}=\cos(84^{\circ})\approx0.1045$, $\sin\theta_{y}=\sin(0^{\circ}) = 0$, $\cos\theta_{y}=\cos(0^{\circ}) = 1$.
$\tan\theta=\frac{7\times0.9945+4\times0}{7\times0.1045 + 4\times1}=\frac{6.9615}{4.7315}\approx1.4713$.
Then $\theta=\arctan(1.4713)\approx55.869^{\circ}$.

Answer:

$|\vec{x}+\vec{y}|\approx8.417$, $\theta\approx55.870^{\circ}$