Question

In Exercises 39–42, let u = -i + j, v = 3i - 2j, and w = -5j. Find each specified scalar or vector.projᵤ (v + w)

Accepted Answer

First, find the vector sum $$ \mathbf{v} + \mathbf{w} . $$Given $$ \mathbf{v} = 3\mathbf{i} - 2\mathbf{j} $$ and $$ \mathbf{w} = -5\mathbf{j} $$, add the corresponding components: $$ \mathbf{v} + \mathbf{w} = (3\mathbf{i} - 2\mathbf{j}) + (0\mathbf{i} - 5\mathbf{j}) = 3\mathbf{i} + (-2 - 5)\mathbf{j} = 3\mathbf{i} - 7\mathbf{j} . $$Recall the formula for the projection of a vector $$ \mathbf{a} $$ onto another vector $$ \mathbf{b} $$:

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

Here, $$ \mathbf{a} = \mathbf{v} + \mathbf{w} $$ and $$ \mathbf{b} = \mathbf{u} = -\mathbf{i} + \mathbf{j} . $$Calculate the dot product $$ (\mathbf{v} + \mathbf{w}) \cdot \mathbf{u} . $$Using the components:

$$ (3, -7) \cdot (-1, 1) = 3 	imes (-1) + (-7) 	imes 1 = -3 - 7 = -10 $$ Calculate the dot product $$ \mathbf{u} \cdot \mathbf{u}  to $$find the denominator:

$$ (-1, 1) \cdot (-1, 1) = (-1)^2 + 1^2 = 1 + 1 = 2 $$ Substitute these values into the projection formula:

$$ 	ext{proj}_{\mathbf{u}} (\mathbf{v} + \mathbf{w}) = \left( \frac{-10}{2} ight) \mathbf{u} = -5 \mathbf{u} $$

This means multiply each component of $$ \mathbf{u}  by$$ $$ -5  to $$get the projection vector.

In Exercises 39–42, let u = -i + j, v = 3i - 2j, and w = -5j. Find each specified scalar or vector.
projᵤ (v + w)

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Key Concepts

Vector Addition

Vector Projection

Dot Product of Vectors