Let vector u, vector v and vector w be vectors in R^3such that vector u + vector v + vector w = vector 0. Show that (vector u) cross product (vector v) = (vector v) cross product (vector w).

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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The problem provides the information that the vectors `bar u, bar v, bar w` are vectors in ` R^3` , such that:

`bar u = x_u bar i + y_u bar j + z_u bar k`

`bar v = x_v bar i + y_v bar j + z_v bar k`

`bar w = x_w bar i + y_w bar j + z_w bar k`

The problem provides the information that the summation of vectors yields `bar 0` , such that:

`bar u + bar v + bar w = bar 0 => {(x_u + x_v + x_w = 0),(y_u + y_v + y_w = 0),(z_u + z_v + z_w = 0):}`

You need to test if the cros product `bar u x bar v = bar v x bar w` using formal determinant, such that:

`bar u x bar v = [(bar i,bar j,bar k),(x_u,y_u,z_u),(x_v,y_v,z_v)]`

`bar u x bar v = y_uz_v bar i + x_uy_v bar k + x_vz_u bar j - x_vy_u bar k - y_vz_u bar i - x_uz_vbar j`

`bar u x bar v = (y_uz_v - y_vz_u) bar i + (x_vz_u - x_uz_v) bar j + (x_uy_v - x_vy_u) bar k`

`bar v x bar w = [(bar i,bar j,bar k),(x_v,y_v,z_v),(x_w,y_w,z_w)]`

`bar v x bar w = y_vz_wbar i + x_vy_w bar k + x_wz_v bar j - x_wy_v bar k - y_wz_v bar i - x_vz_w bar j`

`bar v x bar w = (y_vz_w - y_wz_v)bar i + (x_wz_v - x_vz_w) bar j + (x_vy_w - x_wy_v) bar k`

You need to test if `(y_uz_v - y_vz_u) bar i + (x_vz_u - x_uz_v) bar j + (x_uy_v - x_vy_u) bar k = (y_vz_w - y_wz_v)bar i + (x_wz_v - x_vz_w) bar j + (x_vy_w - x_wy_v) bar k `

`=> {(y_uz_v - y_vz_u = y_vz_w - y_wz_v),(x_vz_u - x_uz_v = x_wz_v - x_vz_w),(x_uy_v - x_vy_u = x_vy_w - x_wy_v):} `

`=> {(y_uz_v +y_wz_v = y_vz_w + y_vz_u),(x_vz_u +x_vz_w = x_wz_v +x_uz_v),(x_uy_v +x_wy_v = x_vy_w +x_vy_u ):} `

`=> {(z_v(y_u + y_w) = y_v(z_w + z_u)),(x_v(z_u + z_w) = z_v(x_w +x_u)),(y_v(x_u +x_w) = x_v(y_w + y_u)):} `

`=> {(z_v(-y_v) = y_v(-z_v)),(x_v(-z_v) = z_v(-x_v)),(y_v(-x_v) = x_v(-y_v)):} `

`=> {(-z_v(-y_v) + z_v(-y_v) = 0 ),(-x_v(z_v) + x_v(z_v) = 0),(-y_v(x_v) + y_v(x_v)= 0):} `

`=> {(0 = 0 ),(0 = 0),( 0 = 0):}` valid

Hence, testing if the cross products` bar u x bar v`   and `bar v x bar w` are equal, using formal determinant, yields that the statement holds.

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