Question

Calculate the angle between (2,2,4) and (2,-1,1). The result is a familiar angle, so the answer is to be given exactly.

Answer 1

Answer:

The angle between $(2,2,4)$ and $(2,-1,1)$ is 60º.

Step-by-step explanation:

From linear algebra, we can determine the angle between both vectors by definition of dot point:

$\cos \theta = \frac{\vec u\bullet \vec v}{\|\vec u\|\cdot \| \vec v\|}$ (1)

Where:

$\vec u$, $\vec v$ - Vectors.

$\|\vec u\|$, $\|\vec v\|$ - Norms of vectors.

$\theta$ - Angle between vectors, measured in sexagesimal degrees.

If we know that $\vec u = (2,2,4)$ and $\vec v = (2,-1,1)$, then angle between vectors is:

$\|\vec u\| = \sqrt{\vec u\bullet \vec u}$ (2)

$\|\vec u\| = \sqrt{2^{2}+2^{2}+4^{2}}$

$\|\vec u\| \approx 4.899$

$\|\vec v\| = \sqrt{\vec v\bullet \vec v}$ (3)

$\|\vec v\| = \sqrt{2^{2}+(-1)^{2}+1^{2}}$

$\|\vec v\| \approx 2.450$

$\vec u \bullet \vec v = (2,2,4)\bullet (2,-1,1)$

$\vec u \bullet \vec v = 4-2+4$

$\vec u \bullet \vec v = 6$

$\cos \theta = \frac{6}{(4.899)\cdot (2.450)}$

$\cos \theta = 0.5$

$\theta = 60^{\circ}$

The angle between $(2,2,4)$ and $(2,-1,1)$ is 60º.