Question

Use the equation a = IaIâ

to express each of the following vectors as the product of its magnitude and (unit vector) direction. (Your instructors prefer angle bracket notation < > for vectors.)
(a) (2, 1, -3)
(b) 2i - 3j + 4k
(c) the sum of (1, 2, -3) and (2, 4, 1)

Answer 1

a)
`\:<\:2,1,-3\:>\:=√(14)\cdot (\:<\:2,1,-3\:>\:)/(√(14) )`

b)
`\:<\:2,-3,4\:>\:=√(29) \cdot (\:<\:2,-3,4\:>\:)/(√(29) )`

c)
`\:<\:3,6,-2\:>\:=7\cdot (\:<\:3,6,-2\:>\:)/(7)`

Explanation:

a) Let a=<2,1,-3>

The magnitude of a is
`|a|=√(2^2+1^2+(-3)^2)`

`|a|=√(4+1+9)=√(14)`

The unit vector in the direction of a is

`\hat{a}=(\:<\:2,1,-3\:>\:)/(√(14) )`

Using the relation
`a=|a|\hat{a}`, we have

`\:<\:2,1,-3\:>\:=√(14)\cdot (\:<\:2,1,-3\:>\:)/(√(14) )`

b) Let a=2i - 3j + 4k

`|a|=√(2^2+(-3)^2+4^2)`

`|a|=√(4+9+16)=√(29)`

`\hat{a}=(\:<\:2,-3,4\:>\:)/(√(29) )`

Using the relation
`a=|a|\hat{a}`, we have

`\:<\:2,-3,4\:>\:=√(29) \cdot (\:<\:2,-3,4\:>\:)/(√(29) )`

c) Let us first find the sum of <1, 2, -3> and <2, 4, 1> to get:

<1+2, 2+4, -3+1>=<3, 6, -2>

Let a=<3, 6, -2>

The magnitude is

`|a|=√(3^2+6^2+(-2)^2)`

`|a|=√(9+36+4)=√(49)=7`

The unit vector in the direction of a is

`\hat{a}=(\:<\:3,6,-2\:>\:)/(7)`

Using the relation
`a=|a|\hat{a}`, we have

`\:<\:3,6,-2\:>\:=7\cdot (\:<\:3,6,-2\:>\:)/(7)`