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Find the orthogonal projections of the vectors $\mathbf{e}_{1}=(1,0)$ and $\mathbf{e}_{2}=(0,1)$ on the line $L$ that makes an angle $\theta$ with the positive $x$-axis in $R^{2}$.
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$\mathbf{a}=(\cos \theta, \sin \theta)$ is a unit vector along the line $L$, so our first problem is to find the orthogonal projection of $\mathbf{e}_{1}$ along a. Since $$\|\mathbf{a}\|=\sqrt{\sin ^{2} \theta+\cos ^{2} \theta}=1 \text { and } \mathbf{e}_{1} \cdot \mathbf{a}=(1,0) \cdot(\cos \theta, \sin \theta)=\cos \theta$$ it follows that this projection is $$\operatorname{proj}_{\mathbf{a}} \mathbf{e}_{1}=\frac{\mathbf{e}_{1} \cdot \mathbf{a}}{\|\mathbf{a}\|^{2}} \mathbf{a}=(\cos \theta)(\cos \theta, \sin \theta)=\left(\cos ^{2} \theta, \sin \theta \cos \theta\right)$$ Similarly, since $\mathbf{e}_{2} \cdot \mathbf{a}=(0,1) \cdot(\cos \theta, \sin \theta)=\sin \theta$, it follows that $$\operatorname{proj}_{\mathbf{a}} \mathbf{e}_{2}=\frac{\mathbf{e}_{2} \cdot \mathbf{a}}{\|\mathbf{a}\|^{2}} \mathbf{a}=(\sin \theta)(\cos \theta, \sin \theta)=\left(\sin \theta \cos \theta, \sin ^{2} \theta\right)$$
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