If the angle between the vectors A⃗ and B⃗ is θ, the value of the product. (B⃗ ×A⃗) A⃗ equals (A

step 1 In this question we will learn about some basic concept of the vectors. The question is if the a

If the angle between the vectors A⃗ and B⃗ is θ, the value...

If the angle between the vectors $\vec{A}$ and $\vec{B}$ is $\theta$, the value of the product. $(\vec{B} \times \vec{A}) \vec{A}$ equals
(A) $B A^{2} \sin \theta$
(B) $B A^{2} \cos \theta \sin \theta$
(C) $B A^{2} \cos \theta$
(D) Zero

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Key Concepts

Cross Product

The cross product is an operation on two vectors in three-dimensional space that results in a third vector, which by definition is perpendicular to the plane containing the original vectors. The magnitude of the cross product is given by the product of the magnitudes of the two vectors multiplied by the sine of the angle between them. This property of perpendicularity is essential when analyzing vector operations involving cross products.

Dot Product

The dot product is an operation that takes two vectors and returns a scalar. It is defined as the product of the magnitudes of the two vectors and the cosine of the angle between them. In the context of a cross product output, taking the dot product with one of the original vectors leads to zero, due to the perpendicular (orthogonal) relation between the cross product result and the original vector.

Orthogonality

Orthogonality refers to the condition where two vectors are perpendicular to each other. In vector operations, particularly with the cross product, the resulting vector is always orthogonal to the original vectors. This key property explains why performing a dot product between one of the original vectors and the cross product yields a result of zero, since the cosine of 90° is zero.

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