Three vectors 𝐀, 𝐁 and 𝐂 satisfy the relation 𝐀 𝐁=0 and 𝐀 𝐂=0 . If 𝐁 and 𝐂 are not lying in the

step 1 In this question we have given three vectors that is A, B and C and they satisfy the relation th

Three vectors 𝐀, 𝐁 and 𝐂 satisfy the relation 𝐀 𝐁=0 and 𝐀...

Three vectors $\mathbf{A}, \mathbf{B}$ and $\mathbf{C}$ satisfy the relation $\mathbf{A B}=0$ and $\mathbf{A C}=0 .$ If $\mathbf{B}$ and $\mathbf{C}$ are not lying in the same plane, then $\mathbf{A}$ is parallel to (a) $\mathrm{B}$ (b) $\mathrm{C}$ (c) $\mathrm{B} \times \mathrm{C}$ (d) $\mathrm{BC}$

Three vectors $\mathbf{A}, \mathbf{B}$ and $\mathbf{C}$ satisfy the relation $\mathbf{A B}=0$ and $\mathbf{A C}=0 .$ If $\mathbf{B}$ and $\mathbf{C}$ are not lying in the same plane, then $\mathbf{A}$ is parallel to
(a) $\mathrm{B}$
(b) $\mathrm{C}$
(c) $\mathrm{B} \times \mathrm{C}$
(d) $\mathrm{BC}$

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Key Concepts

Vector Plane and Linear Independence

When two vectors are not collinear (or not lying in the same plane with a third vector), they span a plane in three-dimensional space. The orthogonal direction to this plane can be uniquely determined using the cross product. This explains why A must be parallel to B × C if it is orthogonal to both B and C.

Cross Product

The cross product of two non-parallel vectors in three-dimensional space results in a vector that is perpendicular to the plane containing the original vectors. This concept is used to determine the direction of a vector that is orthogonal to a given plane, which in this case is the direction of vector A.

Dot Product

The dot product of two vectors is a scalar that represents the product of their magnitudes and the cosine of the angle between them. When the dot product equals zero, it indicates that the vectors are perpendicular to each other. This concept is crucial for determining orthogonality in vector spaces.

Orthogonality

Orthogonality refers to the condition where two vectors are at right angles to each other. In the context of the problem, the fact that vector A has zero dot product with both B and C implies that A is orthogonal to the plane formed by B and C.

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