The vector which can give unit vector along x-axis with 𝐀=2 𝐢̂-4 𝐣̂+7 𝐤̂, 𝐁=7 𝐢̂+2 𝐣̂-5 𝐤

step 1 So in this question the vector is i cap minus a plus b plus c that equals to i cap minus 2 i cap

The vector which can give unit vector along x-axis with ...

The vector which can give unit vector along $\mathrm{x}$-axis with $\quad \mathbf{A}=2 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+7 \hat{\mathbf{k}}, \quad \mathbf{B}=7 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-5 \mathbf{k} \quad$ and $\mathbf{C}=-4 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ is (a) $4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}$ (b) $-5 \hat{i}-5 \hat{j}+5 \hat{k}$ (c) $-4 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}-5 \hat{\mathrm{k}}$ (d) $4 \hat{i}-5 \hat{j}-5 \hat{k}$

The vector which can give unit vector along $\mathrm{x}$-axis with $\quad \mathbf{A}=2 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+7 \hat{\mathbf{k}}, \quad \mathbf{B}=7 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-5 \mathbf{k} \quad$ and
$\mathbf{C}=-4 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ is
(a) $4 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}$
(b) $-5 \hat{i}-5 \hat{j}+5 \hat{k}$
(c) $-4 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}-5 \hat{\mathrm{k}}$
(d) $4 \hat{i}-5 \hat{j}-5 \hat{k}$

Key Concepts

Reciprocal Basis

In a space spanned by a set of linearly independent vectors, a reciprocal (or dual) basis is defined so that the dot product between a vector from the original basis and one from the reciprocal basis is unity when they are corresponding and zero otherwise. This tool helps in expressing any vector in the space in terms of the original basis, especially in problems requiring the determination of specific components or coefficients.

Triple Scalar Product / Determinants

The triple scalar product of three vectors relates to the volume of the parallelepiped they form and is closely connected with determinants in linear algebra. It is often used to check linear independence of vectors, compute reciprocals in a basis, and solve systems of linear equations involving vector components.

Unit Vector

A unit vector is a vector that has a magnitude of one. In many vector problems, unit vectors are used to indicate pure direction without regard to length. They often form the basis for expressing other vectors or decomposing vectors into components along standard axes, such as the x?axis, y?axis, and z?axis.

Linear Combination of Vectors

Expressing a vector as a linear combination of other vectors means writing it as the sum of those vectors multiplied by appropriate scalar coefficients. This concept is fundamental in understanding how vectors can be represented in terms of a set of basis vectors or any given set of linearly independent vectors.

Please give Ace some feedback