Question

In Exercises 33–38, find projᵥᵥ v. Then decompose v into two vectors, v₁ and v₂, where v₁ is parallel to w and v₂ is orthogonal to w.v = i + 3j, w = -2i + 5j

Accepted Answer

Identify the vectors $$ \mathbf{v} = \mathbf{i} + 3\mathbf{j} $$ and $$ \mathbf{w} = -2\mathbf{i} + 5\mathbf{j} . $$Write them in component form as $$ \mathbf{v} = (1, 3) $$ and $$ \mathbf{w} = (-2, 5) . $$Calculate the projection of $$ \mathbf{v} $$ onto $$ \mathbf{w} $$ using the formula:

$$ 
	ext{proj}_{\mathbf{w}} \mathbf{v} = \left( \frac{\mathbf{v} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}} ight) \mathbf{w} 
$$

where $$ \mathbf{v} \cdot \mathbf{w}  is $$the dot product of $$ \mathbf{v} $$ and $$ \mathbf{w} $$, and $$ \mathbf{w} \cdot \mathbf{w}  is $$the dot product of $$ \mathbf{w} $$ with itself. Compute the dot products:

$$
\mathbf{v} \cdot \mathbf{w} = (1)(-2) + (3)(5) 
$$

and

$$
\mathbf{w} \cdot \mathbf{w} = (-2)^2 + 5^2 
$$ Substitute the dot product values into the projection formula to find $$ 	ext{proj}_{\mathbf{w}} \mathbf{v} . $$This gives you the vector $$ \mathbf{v}_1 $$ which is parallel to $$ \mathbf{w} . $$Find the vector $$ \mathbf{v}_2 $$, which is orthogonal to $$ \mathbf{w} $$, by subtracting the projection from $$ \mathbf{v} $$:

$$
\mathbf{v}_2 = \mathbf{v} - \mathbf{v}_1
$$

This completes the decomposition of $$ \mathbf{v} $$ into $$ \mathbf{v}_1  ($$parallel to $$ \mathbf{w} ) $$and $$ \mathbf{v}_2  ($$orthogonal to $$ \mathbf{w} ).$$

In Exercises 33–38, find projᵥᵥ v. Then decompose v into two vectors, v₁ and v₂, where v₁ is parallel to w and v₂ is orthogonal to w.
v = i + 3j, w = -2i + 5j

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Key Concepts

Vector Projection

Dot Product

Vector Decomposition