Question

1. submit answer get help practice similar v = <9, 8, 4> w = <4, 10, 1> find the cosine of the angle between v and w. cos θ = submit answer next item answers # your answer 01 -

Explanation:

Step1: Recall dot - product formula

The dot - product of two vectors $\mathbf{v}=\langle v_1,v_2,v_3
angle$ and $\mathbf{w}=\langle w_1,w_2,w_3
angle$ is $\mathbf{v}\cdot\mathbf{w}=v_1w_1 + v_2w_2+v_3w_3$, and $\mathbf{v}\cdot\mathbf{w}=\vert\mathbf{v}\vert\vert\mathbf{w}\vert\cos\theta$, so $\cos\theta=\frac{\mathbf{v}\cdot\mathbf{w}}{\vert\mathbf{v}\vert\vert\mathbf{w}\vert}$. First, calculate $\mathbf{v}\cdot\mathbf{w}$.
$\mathbf{v}\cdot\mathbf{w}=(9\times4)+(8\times10)+(4\times1)=36 + 80+4=120$

Step2: Calculate magnitudes of vectors

The magnitude of a vector $\mathbf{v}=\langle v_1,v_2,v_3
angle$ is $\vert\mathbf{v}\vert=\sqrt{v_1^{2}+v_2^{2}+v_3^{2}}$.
$\vert\mathbf{v}\vert=\sqrt{9^{2}+8^{2}+4^{2}}=\sqrt{81 + 64+16}=\sqrt{161}$
$\vert\mathbf{w}\vert=\sqrt{4^{2}+10^{2}+1^{2}}=\sqrt{16 + 100 + 1}=\sqrt{117}$

Step3: Calculate cosine of the angle

$\cos\theta=\frac{\mathbf{v}\cdot\mathbf{w}}{\vert\mathbf{v}\vert\vert\mathbf{w}\vert}=\frac{120}{\sqrt{161}\times\sqrt{117}}=\frac{120}{\sqrt{161\times117}}=\frac{120}{\sqrt{18837}}\approx\frac{120}{137.248}$
$\cos\theta=\frac{120}{\sqrt{18837}}\approx 0.874$

Answer:

$\frac{120}{\sqrt{18837}}$