If a=i+2 j+k, b=i-j+k and 𝐜=𝐢+𝐣-𝐤, A vector in the plane of a and b whose projection on 𝐜 is 1 /

step 1 I have given here three vectors here, A, B, a vector in the plane of A and B, whose projection o

If a=i+2 j+k, b=i-j+k and 𝐜=𝐢+𝐣-𝐤, A vector in the plane of...

If $a=i+2 j+k, b=i-j+k$ and $\mathbf{c}=\mathbf{i}+\mathbf{j}-\mathbf{k}$, A vector in the plane of a and $\mathrm{b}$ whose projection on $\mathbf{c}$ is $1 / \sqrt{3}$, is (a) $4 \mathrm{i}-\mathbf{j}+4 \mathbf{k}$ (b) $3 \mathbf{i}+\mathbf{j}-3 \mathbf{k}$ (c) $2 \mathbf{i}+\mathbf{j}-2 \mathbf{k}$ (d) $4 \mathbf{i}+\mathbf{j}-4 \mathbf{k}$

If $a=i+2 j+k, b=i-j+k$ and
$\mathbf{c}=\mathbf{i}+\mathbf{j}-\mathbf{k}$, A vector in the plane of a and $\mathrm{b}$ whose projection on $\mathbf{c}$ is $1 / \sqrt{3}$, is
(a) $4 \mathrm{i}-\mathbf{j}+4 \mathbf{k}$
(b) $3 \mathbf{i}+\mathbf{j}-3 \mathbf{k}$
(c) $2 \mathbf{i}+\mathbf{j}-2 \mathbf{k}$
(d) $4 \mathbf{i}+\mathbf{j}-4 \mathbf{k}$

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Key Concepts

Unit Vectors and Normalization

A unit vector is one that has a magnitude of one and is used to indicate direction without affecting magnitude. Normalizing a vector (dividing by its magnitude) is often a necessary step when calculating projections, ensuring that the direction is preserved while the scaling is controlled.

Vector Projection

This concept involves finding the component of one vector along the direction of another. It is computed using the dot product of the two vectors divided by the magnitude (or squared magnitude) of the vector being projected onto. This yields a scalar multiple that, when applied to the direction vector, gives the projection.

Span of Vectors

Any two non-parallel vectors in space can define a plane, meaning any vector lying in that plane can be expressed as a linear combination of these two vectors. This concept is crucial for understanding how to represent vectors confined to a given subspace.

Dot Product

The dot product is an operation that multiplies two vectors to yield a scalar. It captures information about the magnitude of the vectors and the cosine of the angle between them, and is fundamental in calculating projections and testing for orthogonality.

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