ABCD is a parallelogram. A, B and C are represented by the position vectors
i + 2j - k, 2i + j -2k and 4i - k respectively. Find:
a) A->D
b) The cosine of <BAD
Hello, scorpion007!
$\displaystyle ABCD$ is a parallelogram.
$\displaystyle A, B$ and $\displaystyle C$ are represented by the position vectors:
. . $\displaystyle i + 2j - k,\;2i + j -2k\;4i - k$, respectively.
$\displaystyle (a)\text{ Find: }\overrightarrow{AD}$
We have: .$\displaystyle A = \langle1,2,\text{-}1\rangle,\;B = \langle2,1,\text{-}2\rangle,\;C = \langle4,0,\text{-}1\rangle,\;D = \langle x,y,z\rangle$
Then: .$\displaystyle \overrightarrow{BA} = \langle\text{-}1,\,1,
\,1\rangle$ and $\displaystyle \overrightarrow{CD} = \langle x-4,\,y-0,\,z+1\rangle$
Since $\displaystyle BA \parallel CD:\;\begin{array}{ccc}x - 4 \,=\,\text{-}1 \\ y\,=\,1 \\ z+1\,=\,1\end{array}$ . $\displaystyle \Rightarrow$ . $\displaystyle \begin{array}{ccc}x=3 \\ y=1 \\ z = 0\end{array}$
Therefore: .$\displaystyle D = \langle3,1,0\rangle\quad\Rightarrow\quad AD \:=\:\langle3-1,\,1-2,\,0-(\text{-}1)\rangle\:=\:\langle2,\text{-}1,1\rangle$
$\displaystyle (b)\text{ Find }\cos(\angle BAD)$
The angle $\displaystyle \theta$ between two vectors $\displaystyle \vec{u}$ and $\displaystyle \vec{v}$ is given by: .$\displaystyle \cos\theta \:=\:\frac{\vec{u}\cdot\vec{v}}{|\vec{u}||\vec{v}| } $
We have: .$\displaystyle \begin{array}{cc}\overrightarrow{AB} = \langle1,\text{-}1,\text{-}1\rangle \\ \overrightarrow{AD} = \langle2,\text{-}1,1\rangle\end{array}$ . $\displaystyle \Rightarrow$ . $\displaystyle \begin{array}{cc}|AB| = \sqrt{3} \\ |AD| = \sqrt{6}\end{array}$
. . and: .$\displaystyle \overrightarrow{AB}\cdot\overrightarrow{AD} \:=\:(1)(2)+(\text{-}1)(\text{-}1)+(\text{-}1)(1) \:=\:2$
Therefore: .$\displaystyle \cos(\angle BAD) \;= \;\frac{2}{\sqrt{3}\sqrt{6}}\;=\;\frac{\sqrt{2}}{3 }$
. . $\displaystyle \left(\theta \:\approx\:61.87^o\right)$
earboth, according to your diagram, isn't a + BC = d or OD? not AD as you have it written?
since AD || BC, and since its a parallelogram, shouldnt AD = BC? so why is AD = a + BC? Im thinking a + BC yields the vector from the origin to D, OD.
EDIT: i see [.math] notation is back.