Show that [`veca` - 2`vecb`] x [a + b] = 3a x b for any arbitrary vectors a and b.If possible, explain in all steps. I have no clue what to do in this question.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to write the vectors `bar a = x_a*bari + y_a barj`  and `bar b =x_b*bari + y_b barj `

You need to evaluate the vector `bar a - 2bar b = x_a*bari + y_a barj - 2x_b*bari- 2y_b barj ` `bar a - 2bar b = (x_a - 2x_b)bar i + (y_a - 2y_b)bar j`

You need to evaluate the vector `bar a +bar b = x_a*bari + y_a barj + x_b*bari+ 2y_b barj ` `bar a + bar b = (x_a + x_b)bar i + (y_a + y_b)bar j`

You need to multiply the vectors bar a - 2bar b and bar a + bar b  such that:

`(bar a - 2bar b)*(bar a + bar b) = (x_a - 2x_b)(x_a + x_b) + (y_a - 2y_b)(y_a + y_b)`

You need to remove the brackets such that:

`(bar a - 2bar b)*(bar a + bar b) = x^2_a + x_a*x_b - 2x_a*x_b - 2x^2_b + y^2_a + y_a*y_b - 2y_a*y_b - 2y^2_b `

`(bar a - 2bar b)*(bar a + bar b) = x^2_a - x_a*x_b - 2x^2_b + y^2_a - y_a*y_b - 2y^2_b`

You need to evaluate `3bar a x bar b = (3x_a*bari + 3y_a barj)(x_b*bari + y_b barj )` `3bar a x bar b = 3x_a*x_b + 3y_a*y_b`

Notice that `x^2_a - x_a*x_b - 2x^2_b + y^2_a - y_a*y_b - 2y^2_b != 3x_a*x_b + 3y_a*y_b` , hence `(bar a - 2bar b)*(bar a + bar b) != 3bar a* bar b .`

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