Projection of a=2 i-3 j+k along a vector perpendicular to b=i+3 j-k has the absolute value. (a)

step 1 In this question two vectors are given and we have to find projection of A along a vector perpen

Projection of a=2 i-3 j+k along a vector perpendicular to...

Projection of $\overline{\mathrm{a}}=2 \overline{\mathrm{i}}-3 \overline{\mathrm{j}}+\overline{\mathrm{k}}$ along a vector perpendicular to $\overline{\mathrm{b}}=\overline{\mathrm{i}}+3 \overline{\mathrm{j}}-\overline{\mathrm{k}}$ has the absolute value. (a) $\sqrt{\frac{10}{17}}$ (b) $3 \sqrt{\frac{10}{11}}$ (b) $2 \sqrt{\frac{10}{11}}$ (d) $\sqrt{\frac{10}{11}}$

Projection of $\overline{\mathrm{a}}=2 \overline{\mathrm{i}}-3 \overline{\mathrm{j}}+\overline{\mathrm{k}}$ along a vector perpendicular to $\overline{\mathrm{b}}=\overline{\mathrm{i}}+3 \overline{\mathrm{j}}-\overline{\mathrm{k}}$ has the absolute value.
(a) $\sqrt{\frac{10}{17}}$
(b) $3 \sqrt{\frac{10}{11}}$
(b) $2 \sqrt{\frac{10}{11}}$
(d) $\sqrt{\frac{10}{11}}$

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Key Concepts

Vector Projection

Vector projection is the process of determining the component of one vector along the direction of another vector. It involves calculating how much of the original vector is in the direction of the target vector. This concept is pivotal when breaking down vectors into parts that are parallel and perpendicular to a specific direction.

Orthogonal Decomposition

Orthogonal decomposition is the method of splitting a vector into two components—one that is parallel to a given vector and another that is perpendicular to it. This is particularly useful in many applications of vector analysis since it enables the study of effects acting along different independent directions.

Dot Product

The dot product is a fundamental operation in vector algebra that measures the product of the magnitudes of two vectors and the cosine of the angle between them. It is extensively used to compute projections, as it directly relates to finding the scalar component of one vector in the direction of another.

Magnitude of a Vector

The magnitude (or absolute value) of a vector represents its length and is computed as the square root of the sum of the squares of its components. In the context of projections, the magnitude of the projected component indicates the 'size' of the vector in that particular direction.

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