# Let vector x =< 0,2,-3> and vector y= <0,2,-3> Find vector w such that |w|= 7(|x|) + |y|.

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### 1 Answer

You need to evaluate the lengths of the vectors `bar x` and `bar y` , such that:

`|bar x| = sqrt(0^2 + 2^2 + (-3)^2) => |bar x| = sqrt13`

`|bar y| = sqrt((0^2 + 2^2 + (-3)^2) => |bar y| = sqrt13`

The problem provides the information that `|bar w| = 7|bar x| + |bar y|` such that:

`|bar w| = 7sqrt 13 + sqrt 13`

You need to write the vector `bar w` such that:

`bar w = a*bar i + b*bar j + c*bar k`

`|bar w| = sqrt(a^2 + b^2 + c^2)`

`sqrt(a^2 + b^2 + c^2) = 8 sqrt 13`

`a^2 + b^2 + c^2 = 64*13 = 832`

**Since there are three unknowns to determine, a,b and c and since the problem provides only one information `|bar w| = 7|bar x| + |bar y|` , the vector bar w cannot be fully determined, hence, the only valid condition the vector `bar w` needs to carry out is `a^2 + b^2 + c^2 = 832` .**