If a=i+j-k, b=i-j+k and c is a unit vector perpendicular to...

If $a=i+j-k, b=i-j+k$ and $c$ is a unit vector perpendicular to the vector $a$ and coplanar with $a$ and $b$, then a unit vector $d$ perpendicular to both $a$ and $c$ is (A) $\frac{1}{\sqrt{6}}(2 i-j+k)$ (B) $\frac{1}{\sqrt{2}}(i+j)$ (C) $\frac{1}{\sqrt{2}}(j+k)$ (D) $\frac{1}{\sqrt{2}}(i+k)$

If $a=i+j-k, b=i-j+k$ and $c$ is a unit vector perpendicular to the vector $a$ and coplanar with $a$ and $b$, then a unit vector $d$ perpendicular to both $a$ and $c$ is
(A) $\frac{1}{\sqrt{6}}(2 i-j+k)$
(B) $\frac{1}{\sqrt{2}}(i+j)$
(C) $\frac{1}{\sqrt{2}}(j+k)$
(D) $\frac{1}{\sqrt{2}}(i+k)$

Key Concepts

Vector Representation in 3D Space

This concept refers to the depiction of vectors in three-dimensional space, typically expressed in terms of the standard basis vectors. It provides a method to describe the magnitude and direction of quantities in a structured coordinate system.

Unit Vectors

Unit vectors are vectors with a magnitude of one. They are used to indicate direction without consideration of magnitude and are essential in normalizing vectors during various vector operations, ensuring that comparisons and orthogonal projections are on a common scale.

Orthogonality

Orthogonality in vector spaces means that two vectors are perpendicular to each other, which is mathematically confirmed when their dot product is zero. This concept is crucial in identifying independent directions and in decomposing vectors into mutually perpendicular components.

Coplanarity

Coplanarity describes vectors that lie within the same plane. Identifying coplanar vectors is important when determining projections, rotations, or performing operations that are restricted to a specific two-dimensional plane in three-dimensional space.

Dot Product

The dot product is an algebraic operation that takes two vectors and returns a scalar. It measures the extent to which two vectors align with each other, and a zero dot product indicates that the vectors are orthogonal. This concept is widely used in determining angles and projections between vectors.

Cross Product

The cross product is a vector operation that produces a vector which is perpendicular to the plane formed by two given vectors. It is especially important in calculations involving torque, rotational effects, and in determining a unit vector that is orthogonal to a pair of vectors.

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