# arrow_back Let $\mathbf{u}=(2,-1,3)$ and $\mathbf{a}=(4,-1,2)$. Find the vector component of $\mathbf{u}$ along a and the vector component of $\mathbf{u}$ orthogonal to a.

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Let $\mathbf{u}=(2,-1,3)$ and $\mathbf{a}=(4,-1,2)$. Find the vector component of $\mathbf{u}$ along a and the vector component of $\mathbf{u}$ orthogonal to a.

\begin{aligned} &\mathbf{u} \cdot \mathbf{a}=(2)(4)+(-1)(-1)+(3)(2)=15 \\ &\|\mathbf{a}\|^{2}=4^{2}+(-1)^{2}+2^{2}=21 \end{aligned} Thus the vector component of $\mathbf{u}$ along $\mathbf{a}$ is $$\operatorname{proj}_{\mathbf{a}} \mathbf{u}=\frac{\mathbf{u} \cdot \mathbf{a}}{\|\mathbf{a}\|^{2}} \mathbf{a}=\frac{15}{21}(4,-1,2)=\left(\frac{20}{7},-\frac{5}{7}, \frac{10}{7}\right)$$ and the vector component of $\mathbf{u}$ orthogonal to $\mathbf{a}$ is $$\mathbf{u}-\operatorname{proj}_{\mathbf{a}} \mathbf{u}=(2,-1,3)-\left(\frac{20}{7},-\frac{5}{7}, \frac{10}{7}\right)=\left(-\frac{6}{7},-\frac{2}{7}, \frac{11}{7}\right)$$
Express the vector $\mathbf{u}=(2,3,1,2)$ in the form $\mathbf{u}=\mathbf{w}_{1}+\mathbf{w}_{2}$, where $\mathbf{w}_{1}$ is a scalar multiple of $\mathbf{a}=(-1,0,2,1)$ and $\mathbf{w}_{2}$ is orthogonal to a. Express the vector $\mathbf{u}=(2,3,1,2)$ in the form $\mathbf{u}=\mathbf{w}_{1}+\mathbf{w}_{2}$, where $\mathbf{w}_{1}$ is a scalar multiple of $\m ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 What is the form of vector and parametric equations of lines in$R^{2}$and$R^{3}$1 answer 7 views What is the form of vector and parametric equations of lines in$R^{2}$and$R^{3}$What is the form of vector and parametric equations of lines in$R^{2}$and$R^{3}$... close 1 answer 12 views close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Show that$\mathbf{u}=(-2,3,1,4)$and$\mathbf{v}=(1,2,0,-1)$are orthogonal vectors in$R^{4}$. 1 answer 13 views Show that$\mathbf{u}=(-2,3,1,4)$and$\mathbf{v}=(1,2,0,-1)$are orthogonal vectors in$R^{4}$.Show that$\mathbf{u}=(-2,3,1,4)$and$\mathbf{v}=(1,2,0,-1)$are orthogonal vectors in$R^{4}$. ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Show that$\mathbf{u}=(-2,3,1,4)$and$\mathbf{v}=(1,2,0,-1)$are orthogonal 1 answer 6 views Show that$\mathbf{u}=(-2,3,1,4)$and$\mathbf{v}=(1,2,0,-1)$are orthogonalShow that$\mathbf{u}=(-2,3,1,4)$and$\mathbf{v}=(1,2,0,-1)$are orthogonal ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Find a unit vector that is orthogonal to both$\mathbf{u}=(1,0,1)$and$\mathbf{v}=(0,1,1)$0 answers 9 views Find a unit vector that is orthogonal to both$\mathbf{u}=(1,0,1)$and$\mathbf{v}=(0,1,1)$Find a unit vector that is orthogonal to both$\mathbf{u}=(1,0,1)$and$\mathbf{v}=(0,1,1)$... close 0 answers 6 views Find an initial point$P$of a nonzero vector$\mathbf{u}=\overrightarrow{P Q}$with terminal point$Q(3,0,-5)$and such that$\mathbb{u}$has the same direction as$\mathbf{v}=(4,-2,-1)$.Find an initial point$P$of a nonzero vector$\mathbf{u}=\overrightarrow{P Q}$with terminal point$Q(3,0,-5)$and such that$\mathbb{u}\$ has the sam ...