Question

Find the angle between the vectors. Use a calculator if necessary. (Enter your answer in radians. Round your answer to three decimal places.) (√3,1) and <0, 5 > Find the angle between the vectors. Use a calculator if necessary.<0,4,4> and <3,-3,0>

Answer 1

Answer:

a. The angle $\phi$ between the vectors $\mathbf{u}=\left(\sqrt{3}, 1\right)$ and $\mathbf{v}=\left(0, 5\right)$ is $\phi=\frac{\pi}{3}$ or $60\º$.

b. The angle $\phi$ between the vectors $\mathbf{u}=\left(0, 4, 4\right)$ and $\mathbf{v}=\left(3, -3, 0\right)$ is $\phi=\frac{2 \pi}{3}$ or $120\º$.

Step-by-step explanation:

a. To calculate the angle $\phi$ between the vectors $\mathbf{u}=\left(\sqrt{3}, 1\right)$ and $\mathbf{v}=\left(0, 5\right)$ you must:

Step 1: Calculate the dot product.

The dot product is given as $\displaystyle{\large{{\left({u}_{{x}},{u}_{{y}}\right)}\cdot{\left({v}_{{x}},{v}_{{y}}\right)}={u}_{{x}}\cdot{v}_{{x}}+{u}_{{y}}\cdot{v}_{{y}}}}$.

So,

$\left(\sqrt{3}, 1\right)\cdot\left(0, 5\right)=\left(\sqrt{3}\right)\cdot\left(0\right)+\left(1\right)\cdot\left(5\right)=5$

Step 2: Find the lengths of the vectors.

$\left|\mathbf{u}\right|=\sqrt{\left(u_x\right)^2+\left(u_y\right)^2}=\sqrt{\left(\sqrt{3}\right)^2+\left(1\right)^2}=2$

$\left|\mathbf{v}\right|=\sqrt{\left(v_x\right)^2+\left(v_y\right)^2}=\sqrt{\left(0\right)^2+\left(5\right)^2}=5$

Step 3: The angle is given by $\cos\left(\phi\right)=\frac{\mathbf{u} \cdot \mathbf{v}}{\left|\mathbf{u}\right| \cdot \left|\mathbf{v}\right|}$

$\frac{5}{2 \cdot 5}=\frac{1}{2}$

$\phi=cos^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{3}=60^0$

b. To calculate the angle $\phi$ between the vectors $\mathbf{u}=\left(0, 4, 4\right)$ and $\mathbf{v}=\left(3, -3, 0\right)$ you must:

Step 1: Calculate the dot product.

$\left(0, 4, 4\right)\cdot\left(3, -3, 0\right)=\left(0\right)\cdot\left(3\right)+\left(4\right)\cdot\left(-3\right)+\left(4\right)\cdot\left(0\right)=-12$

Step 2: Find the lengths of the vectors.

$\left|\mathbf{u}\right|=\sqrt{\left(u_x\right)^2+\left(u_y\right)^2+\left(u_z\right)^2}=\sqrt{\left(0\right)^2+\left(4\right)^2+\left(4\right)^2}=4 \sqrt{2}$

$\left|\mathbf{v}\right|=\sqrt{\left(v_x\right)^2+\left(v_y\right)^2+\left(v_z\right)^2}=\sqrt{\left(3\right)^2+\left(-3\right)^2+\left(0\right)^2}=3 \sqrt{2}$

Step 3: The angle is given by $\cos\left(\phi\right)=\frac{\mathbf{u} \cdot \mathbf{v}}{\left|\mathbf{u}\right| \cdot \left|\mathbf{v}\right|}$

$\cos\left(\phi\right)=\frac{\mathbf{u} \cdot \mathbf{v}}{\left|\mathbf{u}\right| \cdot \left|\mathbf{v}\right|}=\frac{-12}{4 \sqrt{2} \cdot 3 \sqrt{2}}=- \frac{1}{2}$

$\phi=cos^{-1}\left(- \frac{1}{2}\right)=\frac{2 \pi}{3}=120^0$