# Assume that (vector v) cross product (vector w) = < 1, 2, -2 >. Evaluate (2(vector v) + 4(vector w)) cross product (-2(vector v) +(vector w)).

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### 1 Answer

You need to write the vectors `bar w` and `bar v` such that:

`bar v = v_x bar i + v_y bar j + v_z bar k`

`bar w = w_x bar i + w_y bar j + w_z bar k`

Evauating the cross product of vectors, yields:

`bar v x bar w = [(bar i, bar j, bar k),(v_x, v_y, v_z),(w_x, w_y, w_z)]`

The problem provides the information that `bar v x bar w = bar i + 2 bar j - 2 bar k` , such that:

`(v_y*w_z - v_z*w_y) bar i + (v_z*w_x - v_x*w_z) bar j + (v_x*w_y - v_y*w_x) bar k = bar i + 2 bar j - 2 bar k`

Equating the coefficients of the same unit vectors, yields:

`v_y*w_z - v_z*w_y = 1`

`v_z*w_x - v_x*w_z = 2`

`v_x*w_y - v_y*w_x = -2`

You need to evaluate the cross product `(2bar v + 4bar w)x(-2 bar v + bar w)` such that:

`2bar v + 4bar w = (2v_x + 4w_x)bar i + (2v_y + 4w_y)bar j + (2v_z + 4w_z) bar k`

`bar w - 2 bar v = (w_x - 2v_x)bar i + (w_y - 2v_y)bar j + (w_z - 2v_z) bar k`

`(2bar v + 4bar w)x(-2 bar v + bar w) = [(bar i, bar j, bar k),(2v_x + 4w_x,2v_y + 4w_y,2v_z + 4w_z),(w_x - 2v_x,w_y - 2v_y,w_z - 2v_z)]`

`(2bar v + 4bar w)x(-2 bar v + bar w) = (2v_y + 4w_y)(w_z - 2v_z)bar i + (2v_z + 4w_z)(w_x - 2v_x)bar j + (2v_x + 4w_x)(w_y - 2v_y) bar k - (w_x - 2v_x)(2v_y + 4w_y)bar k - (w_y - 2v_y)(2v_z + 4w_z)bar i - (2v_x + 4w_x) (w_z - 2v_z)bar j`

`(2bar v + 4bar w)x(-2 bar v + bar w) = (2v_yw_z - 4v_yv_z + 4w_yw_z - 8w_yv_z - 2w_yv_y - 4w_yw_z + 4v_yv_z + 8v_yw_z)bar i + (2v_zw_x - 4v_xv_y + 4w_xw_z - 8v_xw_z - 2v_xw_z + 4v_xv_z - 4w_xw_z + 8v_zw_x)bar j + (2v_xw_y - 4v_xv_y + 4w_xw_y - 8v_yw_x - 2v_yw_x - 4w_xw_y + 4v_xv_y + 8v_xw_y)bar k`

`(2bar v + 4bar w)x(-2 bar v + bar w) = (10v_yw_z - 10w_yv_z)bar i + (10v_zw_x - 10v_xw_z)bar j + (10v_xw_y - 10v_yw_x)bar k`

Factoring out 10 yields:

`(2bar v + 4bar w)x(-2 bar v + bar w) = 10(v_yw_z - w_yv_z)bar i + 10(v_zw_x - v_xw_z)bar j + 10(v_xw_y - v_yw_x)bar k`

Substituting 1 for `v_y*w_z - v_z*w_y` , 2 for `v_z*w_x - v_x*w_z ` and -2 for `v_x*w_y - v_y*w_x` yields:

`(2bar v + 4bar w)x(-2 bar v + bar w) = 10 bar i + 20bar j - 20bar k`

**Hence, evaluating the cross product, under the given conditions, yields `(2bar v + 4bar w)x(-2 bar v + bar w) = 10 bar i + 20bar j - 20bar k` , thus **`(2bar v + 4bar w)x(-2 bar v + bar w) = <10,20,-20>.`