Question

In Exercises 5–8, let v = -5i + 2j and w = 2i - 4j Find the specified vector, scalar, or angle. projᵥᵥv

Accepted Answer

Identify the vectors involved: here, both vectors are $$ \mathbf{v} = -5\mathbf{i} + 2\mathbf{j} $$ and $$ \mathbf{w} = 2\mathbf{i} - 4\mathbf{j} $$, but the problem asks for $$ 	ext{proj}_{\mathbf{v}} \mathbf{v} $$, which means the projection of $$ \mathbf{v} $$ onto itself. Recall the formula for the projection of a vector $$ \mathbf{a} $$ onto a vector $$ \mathbf{b} $$:

$$ 	ext{proj}_{\mathbf{b}} \mathbf{a} = \left( \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} ight) \mathbf{b} $$

where $$ \mathbf{a} \cdot \mathbf{b}  is $$the dot product of $$ \mathbf{a} $$ and $$ \mathbf{b} . $$Since we are projecting $$ \mathbf{v} $$ onto itself, substitute $$ \mathbf{a} = \mathbf{v} $$ and $$ \mathbf{b} = \mathbf{v} $$ into the formula:

$$ 	ext{proj}_{\mathbf{v}} \mathbf{v} = \left( \frac{\mathbf{v} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} ight) \mathbf{v} $$ Calculate the dot product $$ \mathbf{v} \cdot \mathbf{v}  by $$multiplying corresponding components and summing:

$$ \mathbf{v} \cdot \mathbf{v} = (-5)(-5) + (2)(2) $$ Simplify the fraction $$ \frac{\mathbf{v} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} $$ which will equal 1, so the projection $$ 	ext{proj}_{\mathbf{v}} \mathbf{v}  is $$simply $$ \mathbf{v} $$ itself.

In Exercises 5–8, let v = -5i + 2j and w = 2i - 4j Find the specified vector, scalar, or angle. projᵥᵥv

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Key Concepts

Vector Projection

Dot Product of Vectors

Vector Magnitude