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 + 2j, w = 3i + 6j

Accepted Answer

Identify the vectors given: $$ \mathbf{v} = \mathbf{i} + 2\mathbf{j} $$ and $$ \mathbf{w} = 3\mathbf{i} + 6\mathbf{j} . $$Express them in component form as $$ \mathbf{v} = (1, 2) $$ and $$ \mathbf{w} = (3, 6) . $$Recall the formula for the projection of $$ \mathbf{v} $$ onto $$ \mathbf{w} $$:

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

where $$ \mathbf{v} \cdot \mathbf{w}  is $$the dot product of $$ \mathbf{v} $$ and $$ \mathbf{w} . $$Calculate the dot products:

$$
\mathbf{v} \cdot \mathbf{w} = (1)(3) + (2)(6) = 3 + 12
$$

and

$$
\mathbf{w} \cdot \mathbf{w} = (3)^2 + (6)^2 = 9 + 36
$$ Substitute the dot products into the projection formula to find $$ 	ext{proj}_{\mathbf{w}} \mathbf{v} . $$This gives 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 + 2j, w = 3i + 6j

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Key Concepts

Vector Projection

Dot Product of Vectors

Vector Decomposition into Parallel and Orthogonal Components