Final answer:
The unit vector orthogonal to vectors u = <-16, -12, 8> and v = <20, -24, -4> is calculated using their cross product and is <5/6, -1/3, 1/2> after normalizing the cross product to unit length.
Step-by-step explanation:
To find a unit vector orthogonal to both u = <-16, -12, 8> and v = <20, -24, -4>, we need to use the cross product, also known as the vector product. The cross product of two vectors creates a new vector that is perpendicular to both of the original vectors. This can be computed using the determinant of a matrix that includes the unit vectors i, j, and k along with the components of vectors u and v.
First, set up the matrix as follows (ignoring the actual determinant notation for clarity):
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- i j k
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- -16 -12 8
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- 20 -24 -4
Then, calculate the cross product:
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- i-component: (-12 * -4) - (8 * -24) = 48 + 192 = 240
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- j-component: (8 * 20) - (-16 * -4) = 160 - 64 = 96
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- k-component: (-16 * -24) - (-12 * 20) = 384 - 240 = 144
The resulting vector is <240, -96, 144>.
To find the unit vector, divide the resulting vector by its magnitude:
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- Magnitude: √(240² + (-96)² + 144²) = 288
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- Unit vector: <240/288, -96/288, 144/288> = <5/6, -1/3, 1/2>
The final unit vector is <5/6, -1/3, 1/2>, which is orthogonal to both u and v.