<p>Show that the projection vector of $\mathbf{a}$ onto $\mathbf{b}$ is given by $\left( \mathbf{a}\cdot \mathbf{\hat{b}} \right)\mathbf{\hat{b}}$.</p>

Show that the projection vector of $\mathbf{a}$ onto $\mathbf{b}$ is given by $\left( \mathbf{a}\cdot \mathbf{\hat{b}} \right)\mathbf{\hat{b}}$.

Vectors - General

Vectors

H2 Math

International Baccalaureate Higher Level Mathematics

IB HL Math

Show the formula of projection vector of a onto b

When $\mathbf{a} \cdot \mathbf{b} > 0$ (since $\theta$ is acute):

Projection length of $\mathbf{a}$ onto $\mathbf{b} = |\mathbf{a} \cdot \hat{\mathbf{b}}|$

$|\mathbf{a} \cdot \hat{\mathbf{b}}|$ units $\rightarrow |\mathbf{a} \cdot \hat{\mathbf{b}}|\hat{\mathbf{b}}$

Projection vector of $\mathbf{a}$ onto $\mathbf{b} = |\mathbf{a} \cdot \hat{\mathbf{b}}|\hat{\mathbf{b}} = (\mathbf{a} \cdot \hat{\mathbf{b}})\hat{\mathbf{b}}$ since $\mathbf{a} \cdot \hat{\mathbf{b}} > 0$

When $\theta$ is obtuse, $\mathbf{a} \cdot \hat{\mathbf{b}} < 0$:

Projection vector of $\mathbf{a}$ onto $\mathbf{b} = -|\mathbf{a} \cdot \hat{\mathbf{b}}|\hat{\mathbf{b}}$

$= -[-( \mathbf{a} \cdot \hat{\mathbf{b}})]\hat{\mathbf{b}}$ since $\mathbf{a} \cdot \hat{\mathbf{b}} < 0$

$= (\mathbf{a} \cdot \hat{\mathbf{b}})\hat{\mathbf{b}}$

$\therefore$ projection vector of $\mathbf{a}$ onto $\mathbf{b} = (\mathbf{a} \cdot \hat{\mathbf{b}})\hat{\mathbf{b}}$

Show the formula of projection vector of a onto b

Show the formula of projection vector of a onto b

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