Find a unit vector perpendicular to the vectors A = 3i^-4j^ 5k^ and B = i^-j^ k^?

Question

Answer

Finding a Unit Vector Perpendicular to Two Given Vectors

To find a unit vector that is perpendicular to two given vectors A and B, we need to use the cross product of the two vectors. The cross product of two vectors gives us a vector that is perpendicular to both of them.

Cross Product of Two Vectors

The cross product of two vectors A and B is given by:

A ⨯ B = |A| |B| sin θ n

where:

A and B are the two vectors

|A| and |B| are the magnitudes of the two vectors

θ is the angle between the two vectors

n is the unit vector perpendicular to both A and B

Finding the Cross Product of A and B

Let's find the cross product of the two given vectors A and B:

A ⨯ B = (3i^-4j^ 5k^) ⨯ (i^-j^ k^)

= (5i^ - 3j^ - 7k^)

So the vector (5i^ - 3j^ - 7k^) is perpendicular to both A and B.

Find the Unit Vector Perpendicular to A and B

Now we need to find the unit vector perpendicular to both A and B. We can do that by dividing the vector (5i^ - 3j^ - 7k^) by its magnitude:

|n| = √(5^2 + (-3)^2 + (-7)^2) = √83

n = (5i^ - 3j^ - 7k^) / √83

So the unit vector perpendicular to both A and B is:

n = (5/√83)i^ - (3/√83)j^ - (7/√83)k^

This is the required unit vector.

Answer

Find their cross product and divide it by its magnitude.
(A vector perpendicular to two vectors is given by their cross product. )

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