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gruba
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Homework Statement
Given points of a triangle: [itex]A(4,1,-2),B(2,0,0),C(-2,3,-5)[/itex]. Line [itex]p[/itex] contains point [itex]B[/itex], is orthogonal to [itex]\overline{AC}[/itex], and is coplanar with [itex]ABC[/itex]. Intersection of [itex]p[/itex] and [itex]\overline{AC}[/itex] is the point [itex]B_1[/itex].
Find vector [itex]\overrightarrow{B_1B}[/itex].
Homework Equations
-Vector projection
- Dot product
-Magnitude of a vector
The Attempt at a Solution
[tex]proj_{\overrightarrow{AC}}\overrightarrow{AB}=\overrightarrow{AB_1}=\frac{\overrightarrow{AB}\cdot \overrightarrow{AC}}{|\overrightarrow{AC}|^2}\cdot \overrightarrow{AC}[/tex]
[tex]\overrightarrow{AB}=[-2,-1,2],\overrightarrow{AC}=[-6,2,-3],|\overrightarrow{AC}|=7[/tex]
[tex]\overrightarrow{AB}\cdot \overrightarrow{AC}=4[/tex]
[tex]\Rightarrow proj_{\overrightarrow{AC}}\overrightarrow{AB}=\overrightarrow{AB_1}=\left[-\frac{24}{49},\frac{8}{49},-\frac{12}{49}\right][/tex]
From [itex]\overrightarrow{AB_1}[/itex] we can find the point [itex]B_1\Rightarrow B_1=\left(\frac{172}{49},\frac{8}{49},-\frac{-110}{49}\right)[/itex] [tex]\Rightarrow \overrightarrow{B_1B}=\left[-\frac{74}{49},-\frac{57}{49},\frac{110}{49}\right][/tex]
Is this correct?
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