# Assume that (vector u cross prodcut vector v) = <38,-34,-24>.Use the properties of the cross-product to evaluate the following expression. (-1(vector u)+(vector v)) cross product (4(vector...

Assume that (vector u cross prodcut vector v) = <38,-34,-24>.

Use the properties of the cross-product to evaluate the following expression.

(-1(vector u)+(vector v)) cross product (4(vector u)+4(vector v))

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You need to consider `bar u = u_x bar i + u_y bar j + u_z bar k` and `bar v = v_x bar i + v_y bar j + v_z bar k` .

Evaluating `bar v - bar u` yields:

`bar v - bar u = (v_x - u_x) bar i + (v_y - u_y) bar j + (v_z - u_z) bar k`

Evaluating `4bar v + 4bar u` yields:

`4bar v + 4bar u = 4(bar v + bar u) = 4(v_x + u_x) bar i + 4(v_y + u_y) bar j + 4(v_z + u_z) bar k`

Assuming that the vector `bar v - bar u` yields `bar p` and `4bar v + 4bar u` yields `bar q` , you need to use the formula of cross product such that:

`bar p x bar q = [(bar i, bar j, bar k),(v_x - u_x , v_y - u_y , v_z - u_z),(4v_x + 4u_x , 4v_y + 4u_y , 4v_z + 4u_z)]`

`bar p x bar q = 4(v_y - u_y)(v_z + u_z) bar i + 4(v_x + u_x)(v_x - u_z) bar j + 4(v_y + u_y)(v_x - u_x) bar k - 4(v_x + u_x)(v_y - u_y) bar k - 4(v_y + u_y)(v_z - u_z) bar i - 4(v_z + u_z)(v_x - u_x) bar j`

`bar p x bar q = (4(v_y - u_y)(v_z + u_z) - 4(v_y + u_y)(v_z - u_z))bar i + (4(v_x + u_x)(v_x - u_z) - 4(v_z + u_z)(v_x - u_x)) bar j + (4(v_y + u_y)(v_x - u_x) - 4(v_x + u_x)(v_y - u_y))`

Since the problem provides the information that `bar p x bar q = <38,-34,-24>` yields:

`4(v_y - u_y)(v_z + u_z) - 4(v_y + u_y)(v_z - u_z) = 38`

`4(v_x + u_x)(v_x - u_z) - 4(v_z + u_z)(v_x - u_x) = -34`

`4(v_y + u_y)(v_x - u_x) - 4(v_x + u_x)(v_y - u_y) = -24`

**Hence, using the properties of vector operations yields `4(v_y - u_y)(v_z + u_z) - 4(v_y + u_y)(v_z - u_z) = 38 ; 4(v_x + u_x)(v_x - u_z) - 4(v_z + u_z)(v_x - u_x) = -34` and `(v_y + u_y)(v_x - u_x) - (v_x + u_x)(v_y - u_y) = -6` .**