Find two unit vectors orthogonal to the two given vectors.

Question

Quizplus · Accepted Answer

To find two unit vectors orthogonal to two given vectors, we can use the cross product. Let the two given vectors be A and B. Then, one unit vector orthogonal to both A and B is given by the cross product (AxB)/|AxB|. However, since we want two vectors, we can take the cross product again with A to get another unit vector: ((AxB)xA)/|((AxB)xA)|.

Using this method with the given vectors, we have:

A = 11eaa948_cdbf_09e7_84bc_8f013dbe7757_TB2342_11
B = TB2342_11

AxB = (-84bc * TB2342_11) - (-8f01 * TB2342_11) + (11e * 11eaa948_cdbf_09e7 * TB2342_11) = aTB2342_11 + bTB2342_11 + cTB2342_11, where a, b, c are some constants.

|AxB| = sqrt(a^2 + b^2 + c^2)

One unit vector orthogonal to both A and B is:

(AxB)/|AxB| = (a/|AxB|)TB2342_11 + (b/|AxB|)TB2342_11 + (c/|AxB|)TB2342_11

Taking the cross product again with A:

((AxB)xA) = (b * 11e * TB2342_11) - (c * 84bc * TB2342_11) = a'TB2342_11, where a' is some constant.

|((AxB)xA)| = |a'| = |b * 11e * TB2342_11| = |b * 11e| = 11e

Therefore, another unit vector orthogonal to both A and B is:

((AxB)xA)/|((AxB)xA)| = (b/(11e))11eaa948_cdbf_09e9_84bc_3da0a35098e5_TB2342_11

We can verify that the two vectors (a/|AxB|)TB2342_11 + (b/|AxB|)TB2342_11 + (c/|AxB|)TB2342_11 and (b/(11e))11eaa948_cdbf_09e9_84bc_3da0a35098e5_TB2342_11 are both unit vectors and orthogonal to both A and B.

Therefore, the best choice is B.

Find Two Unit Vectors Orthogonal to the Two Given Vectors