# arrow_back What is the form of vector and parametric equations of lines in $R^{2}$ and $R^{3}$

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What is the form of vector and parametric equations of lines in $R^{2}$ and $R^{3}$

The variable $t$ (called a parameter) varies from $-\infty$ to $\infty$, the point $\mathbf{x}$ traces out the line $L$.
Accordingly, we have the following result. Let $L$ be the line in $R^{2}$ or $R^{3}$ that contains the point $\mathbf{x}_{0}$ and is parallel to the nonzero vector $\mathbf{v}$.
Then the equation of the line through $\mathbf{x}_{0}$ that is parallel to $\mathbf{v}$ is $$\mathbf{x}=\mathbf{x}_{0}+t \mathbf{v}$$ If $\mathbf{x}_{0}=\mathbf{0}$, then the line passes through the origin and the equation has the form $$\mathbf{x}=t \mathbf{v}$$
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 1 answer 12 views close 0 answers 11 views If $\mathbf{n}=(a, b, c)$ is a unit vector, use this formula to show that (perhaps surprisingly) the orthogonal projection of $\mathbf{x}$ into the plane perpendicular to $\mathbf{n}$ is given bya) Let $\mathbf{v}:=(a, b, c)$ and $\mathbf{x}:=(x, y, z)$ be any vectors in $\mathbb{R}^{3}$. Viewed as column vectors, find a $3 \times 3$ m ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Define a vector 1 answer 11 views Define a vectorDefine a vector ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 You may choose the same option for matching more than once. Read through the following proof and spot an error or not in the lines$L 1$to$L 5$. 0 answers 5 views You may choose the same option for matching more than once. Read through the following proof and spot an error or not in the lines$L 1$to$L 5$. You may choose the same option for matching more than once. Read through the following proof and spot an error or not in the lines$L 1$to$L 5$. ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 You may choose the same option for matching more than once. Read through the following proof and match the lines with the reasons given below. 0 answers 9 views You may choose the same option for matching more than once. Read through the following proof and match the lines with the reasons given below. You may choose the same option for matching more than once.&nbsp;Read through the following proof and match the lines with the reasons given below. ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Is it possible to have$\operatorname{proj}_{\mathrm{a}} \mathbf{u}=\operatorname{proj}_{\mathbf{u}} \mathbf{a} ?$Explain. 0 answers 7 views Is it possible to have$\operatorname{proj}_{\mathrm{a}} \mathbf{u}=\operatorname{proj}_{\mathbf{u}} \mathbf{a} ?$Explain.Is it possible to have$\operatorname{proj}_{\mathrm{a}} \mathbf{u}=\operatorname{proj}_{\mathbf{u}} \mathbf{a} ?\$ Explain. ...