What is the angel between vector P and cross product of vectors( P+ Q) and(P-Q)?Here P and Q are vectors.?

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Answer

Angle between vector P and cross product of vectors (P Q) and (P-Q)

Let's begin by understanding the concept of cross product and dot product of vectors.

Cross Product

The cross product of two vectors, say A and B, is a vector that is perpendicular to both A and B. The magnitude of the cross product is the area of the parallelogram formed by the two vectors, and the direction of the cross product is given by the right-hand rule.

Dot Product

The dot product of two vectors, say A and B, is a scalar that is equal to the magnitude of A times the magnitude of B multiplied by the cosine of the angle between A and B. The dot product is also equal to the projection of one vector onto the other multiplied by the magnitude of the vector being projected.

Angle between vectors

The angle between two vectors, say A and B, is given by the formula: cos(theta) = (A.B) / (|A||B|), where A.B is the dot product of A and B, and |A| and |B| are the magnitudes of A and B, respectively.

Angle between vector P and cross product of vectors (P Q) and (P-Q)

Let's denote the cross product of vectors (P Q) and (P-Q) by R. We want to find the angle between vector P and R.

First, we need to find the magnitude of R. Using the formula for the cross product, we get:

R = (P Q) x (P-Q) = PQ x PQ - P(P.Q) + Q(P.Q) - QP x QP = 0 - P(P.Q) + Q(P.Q) - 0 = (Q-P)(P.Q)

The magnitude of R is therefore |R| = |(Q-P)(P.Q)| = |Q-P||P.Q|

Next, we need to find the dot product of P and R. Using the formula for the dot product, we get:

P.R = |P||R|cos(theta)

Rearranging this equation, we get:

cos(theta) = (P.R) / (|P||R|)

Substituting the expressions for P.R and |R| that we found earlier, we get:

cos(theta) = [(P.(Q-P))(P.Q)] / [|P||Q-P||P.Q|]

Simplifying this expression, we get:

cos(theta) = (P.(Q-P)) / [|P||Q-P|]

Therefore, the angle between vector P and R is given by:

theta = cos^-1[(P.(Q-P)) / [|P||Q-P|]]

So, we have now derived the formula for the angle between vector P and the cross product of vectors (P Q) and (P-Q).

Answer

90degree

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