Question

Express each vector as a product of its length and direction.
16–√i−16–√j−16–√k

Answer 1

Question:

Express each vector as a product of its length and direction.

`(1)/(√(6))i - (1)/(√(6))j - (1)/(√(6))k`

Answer:

`(1)/(√(2))`
`((1)/(√(3))i - (1)/(√(3))j - (1)/(√(3))k)`

Explanation:

A vector v can be expressed as a product of its length and direction as follows;

v = |v| u

Where;

|v| = length/magnitude of v

u = unit vector in the direction of v

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Let the given vector be v, i.e

`v = (1)/(√(6))i - (1)/(√(6))j - (1)/(√(6))k`

(i) The length/magnitude |v| of vector v is therefore,

|v| =
`\sqrt{((1)/(√(6)))^2 + (-(1)/(√(6)))^2 + (-(1)/(√(6)))^2`

|v| =
`\sqrt{((1)/(6)) + ((1)/(6)) + ((1)/(6))`

|v| =
`\sqrt{((3)/(6))`

|v| =
`\sqrt{((1)/(2))`

|v| =
`(1)/(√(2))`

(ii) The unit vector u in the direction of vector v, is therefore,

u =
`(v)/(|v|)`

`u = ((1)/(√(6))i - (1)/(√(6))j - (1)/(√(6))k)/((1)/(√(2)))`

`u = √(2)((1)/(√(6))i - (1)/(√(6))j - (1)/(√(6))k)`

`u = ((√(2))/(√(6))i - (√(2))/(√(6))j - (√(2))/(√(6))k)`

`u = ((1)/(√(3))i - (1)/(√(3))j - (1)/(√(3))k)`

Therefore, the vector can be expressed as a product of its length and direction as:

|v| u =
`(1)/(√(2))`
`((1)/(√(3))i - (1)/(√(3))j - (1)/(√(3))k)`