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

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

Explanation:

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

`\cos \theta = (\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\| = √(\vec u\bullet \vec u)` (2)

`\|\vec u\| = \sqrt{2^(2)+2^(2)+4^(2)}`

`\|\vec u\| \approx 4.899`

`\|\vec v\| = √(\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 = (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.