how do i do this please xx
These are very basic practice problems. You will have to show some familiarity with them before you can move on.
Hints:
ai) Dot Product
aii) Cross Product
b)
If Z = a + bi
Z^2 = (a^2 - b^2) + 2abi
Z-bar = a - bi (The conjugate)
c) Factor and use the quadratic formula.
Hello, Amanda!
Here's the first one . . .
We have: .$\displaystyle \begin{array}{ccc}\vec{v} & = & \langle1,\,\text{-}1,\,0\rangle \\ \vec{w} & = & \langle 0,\,1,\,\text{-}2\rangle\end{array}$a) Condider the following position vectors: .$\displaystyle \begin{array}{ccc}\vec{v} & = & i-j\\ \vec{w} & = & j - 2k\end{array}$
(1) Find the angle between the two vectors.
(2) Find a vector perpendicular to both vectors.
(1) We're expected to know this formula: .$\displaystyle \cos\theta \;=\;\frac{|\vec{v}\bullet\vec{w}|}{|\vec{v}||\vec {w}|} $
So we have: .$\displaystyle \cos\theta \;=\;\frac{|(1)(0) + (\text{-}1)(1) + (0)(\text{-}2)|}{\sqrt{1^2+(\text{-}1)^2+0^2}\,\sqrt{0^2+1^2+(\text{-}2)^2}} \;=\;\frac{|0 -1 + 0|}{\sqrt{2}\,\sqrt{5}}\;=\;\frac{1}{\sqrt{10}}$
. . Therefore: .$\displaystyle \theta \;=\;\cos^{-1}\!\left(\frac{1}{\sqrt{10}}\right) \;\approx\;71.6^o$
(2) We're expect to know that a vector perpendicular to two vectors
. . .is the cross product of the two vectors.
We have: .$\displaystyle \vec{n} \;=\;\begin{vmatrix}i & j & k \\ 1 & \text{-}1 & 0 \\ 0 & 1 & \text{-}2\end{vmatrix} \;=\;i(2-0) - j(\text{-}2 -0) + k(1-0) \;=\;2i + 2j + k$
. . Therefore: .$\displaystyle \vec{n} \;=\;\langle 2,\,2,\,1\rangle$
Let $\displaystyle z = 2 - 3j$ and $\displaystyle s = 1 + 2j$
We want:
$\displaystyle \frac{2 + z}{1 + s}$
$\displaystyle = \frac{2 + (2 - 3j)}{1 + (1 + 2j)}$
$\displaystyle = \frac{2 + 2 - 3j}{1 + 1 + 2j}$
$\displaystyle = \frac{4 - 3j}{2 + 2j}$
Now simplify:
$\displaystyle = \frac{4 - 3j}{2 + 2j} \cdot \frac{2 - 2j}{2 - 2j}$
$\displaystyle = \frac{(4 - 3j)(2 - 2j)}{(2 + 2j)(2 - 2j)}$
$\displaystyle = \frac{8 - 14j + 6j^2}{4 - 4j^2}$
$\displaystyle = \frac{8 - 14j - 6}{4 + 4}$
$\displaystyle = \frac{2 - 14j}{8}$
$\displaystyle = \frac{1 - 7j}{4}$
-Dan