Question

Describe all unit vectors orthogonal to both of the given vectors. −4i + 8j − 6k, 6i + 2j + 5k positive i-component=? negative i-component=?

Answer 1

To find unit vectors orthogonal to both given vectors, we can use the cross product. The positive i-component unit vector is 58i, and the negative i-component unit vector is -58i.

Step-by-step explanation:

To find unit vectors orthogonal to both given vectors, we can use the cross product. Let's call the first vector A = -4i + 8j - 6k and the second vector B = 6i + 2j + 5k.

To find the positive i-component unit vector, we can take the cross product of vector A and B as A x B. The resulting vector will have an i-component. Let's perform the cross product:

A x B = (8 * 5 - (-6) * 2)i - ((-4) * 5 - (-6) * 6)j + ((-4) * 2 - 8 * (-6))k

A x B = 58i - 6j + 32k

The positive i-component unit vector = 58i

The negative i-component unit vector will be the negative of the positive i-component unit vector, so the negative i-component = -58i

Answer 2

The unit vector orthogonal to both given vectors is approximately -0.45i + 0.81j - 0.18k. The positive i-component is -0.45, and the negative i-component is 0.45.

Step-by-step explanation:

To find unit vectors orthogonal to both given vectors, we can take their cross product. Let's call the given vectors A and B.

A = -4i + 8j - 6k, B = 6i + 2j + 5k

To find the cross product A × B, we can use the formula for cross product:

A × B = (AyBz - AzBy)i - (AxBz - AzBx)j + (AxBy - AyBx)k

Substituting the values of A and B into the formula, we get:

A × B = (-4)(5)i - (-6)(6)j + (-4)(2)k = -20i + 36j - 8k

Therefore, the unit vector orthogonal to both A and B is (-20i + 36j - 8k)/sqrt((-20)^2 + 36^2 + (-8)^2), which simplifies to approximately -0.45i + 0.81j - 0.18k.

The positive i-component of this unit vector is -0.45, and the negative i-component is 0.45.