# Recent questions tagged planes

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 ...
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Prove the orthogonal projection theorem
Prove the orthogonal projection theoremProve the orthogonal projection theorem ...
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The planes $$x+2 y-2 z=3 \text { and } 2 x+4 y-4 z=7$$ are parallel since their normals, $(1,2,-2)$ and $(2,4,-4)$, are parallel vectors. Find the distance between these planes.The planes $$x+2 y-2 z=3 \text { and } 2 x+4 y-4 z=7$$ are parallel since their normals, $(1,2,-2)$ and $(2,4,-4)$, are parallel vectors. Find the d ...
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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.Is it possible to have $\operatorname{proj}_{\mathrm{a}} \mathbf{u}=\operatorname{proj}_{\mathbf{u}} \mathbf{a} ?$ Explain. ...
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Find the lengths of the sides and the interior angles of the triangle in $R^{4}$ whose vertices are $$P(2,4,2,4,2), \quad Q(6,4,4,4,6), \quad R(5,7,5,7,2)$$
Find the lengths of the sides and the interior angles of the triangle in $R^{4}$ whose vertices are $$P(2,4,2,4,2), \quad Q(6,4,4,4,6), \quad R(5,7,5,7,2)$$Find the lengths of the sides and the interior angles of the triangle in $R^{4}$ whose vertices are  P(2,4,2,4,2), \quad Q(6,4,4,4,6), \quad R(5,7,5 ...
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The angle between vectors $(1,3,-2)$ and $(4,-2,-1)$ is
The angle between vectors $(1,3,-2)$ and $(4,-2,-1)$ isThe angle between vectors $(1,3,-2)$ and $(4,-2,-1)$ is &nbsp; A) 0 B) $\dfrac{\pi}{3}$ C) $\dfrac{\pi}{2}$ D) $\pi$ E) none of the above ...
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The angle between vectors $(2,6,-4)$ and $(4,-2,-1)$ is
The angle between vectors $(2,6,-4)$ and $(4,-2,-1)$ isThe angle between vectors $(2,6,-4)$ and $(4,-2,-1)$ is &nbsp; A. 0 B. $\dfrac{\pi}{3}$ C. $\dfrac{\pi}{2}$ D. $\pi$ E. none of the above ...
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Parametrized Surface ($\underline{r}(u, v)$) and Curve and regular parametrization
Parametrized Surface ($\underline{r}(u, v)$) and Curve and regular parametrization Parametrized Surface ($\underline{r}(u, v)$) and Curve and regular parametrization ...
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Parametrization and Tangents to planes ...
Parametrization and Tangents to planes ...Parametrization and Tangents to planes ... ...
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Given the curve $9x^2+4y^2=36$, identify and then sketch the curve indicating the important points
Given the curve $9x^2+4y^2=36$, identify and then sketch the curve indicating the important pointsGiven the curve $9x^2+4y^2=36$, identify and then sketch the curve indicating the important points ...
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Given $9y^2 + 4z^2 = x^2 + 36$...
Given $9y^2 + 4z^2 = x^2 + 36$... Given $9y^2 + 4z^2 = x^2 + 36$ Rewrite the equation in standard form Identify the surface Draw each trace for the coordinate planes and ...