Answer to Question #113638 in Linear Algebra for kgodiso

Question

consider the vectors;u(1,0)and v(0,1)
questions
1-determine cos theta,where theta is the angle between u and v.
2-determine the area of the parallelogram determined by u and v.


question 2
2.1-Let L1 and L2 be defined by x=W0+su where s is the element of real numbers
and y=w1+tv where t is the element of real numbers .
2.2-find the plane the passes through the point(2,4,-3)and is parallel to the plane -2x+4y-5z+5=0
2.3.find the line that passes through the point (2,5,3)and is perpendicular to the plane 2x-3y+4z+7=0
2.4find an equation of the plane passing through the point(-2,3,4)and is perpendicular to the line passing through the points (4,-2,5)and (0,2,4)

Answer 1

$\vec{u}=(1,0),\vec{v}=(0,1)$

1.1

$\vec{u}\vec{v}=|\vec{u}||\vec{v}|\cos\theta\\ \cos\theta=\frac{\vec{u}\vec{v}}{|\vec{u}||\vec{v}|}\\ \vec{u}\vec{v}=1\cdot0+0\cdot1=0\\ \cos\theta=0$

1.2

$S=|\vec{u}||\vec{v}|\sin\theta\\ |\vec{u}|=\sqrt{1^2+0^2}=1\\ |\vec{v}|=\sqrt{0^2+1^2}=1\\ \cos\theta=0, \theta=90^0\\ S=1\cdot1\cdot\sin90^0=1$

2.1

$L_1:x=w_0+su,s\in R\\ w_0=(w_{00},w_{01}), u=(a,b)\\ x=w_{00}+as\\ y=w_{01}+bs,s\in R\\ s=\frac{x-w_{00}}{a}\\ y=w_{01}+b\cdot\frac{x-w_{00}}{a}\\ k=\frac{b}{a}\\ L_1:x=w_1+tv,t\in R\\ w_1=(w_{10},w_{11}), v=(c,d)\\ x=w_{10}+tc\\ y=w_{11}+td,t\in R\\ t=\frac{x-w_{10}}{c}\\ y=w_{11}+d\cdot\frac{x-w_{10}}{c}\\ k_1=\frac{d}{c}\\ L_1||L_2:\\ k=pk_1\\ \frac{b}{a}=p\cdot\frac{d}{c}$

2.2

$\alpha:-2x+4y-5z+5=0\\ \vec{n}=(-2,4,-5)\\ \beta: A(2,4,-3)\in \beta, \vec{n}||\beta\\ -2(x-2)+4(y-4)-5(z+3)=0\\ -2x+4y-5z-27=0$

2.3

$\alpha:2x-3y+4z+7=0\\ \vec{n}=(2,-3,4) \bot\alpha\\ \alpha||a, \vec{n}\bot a, A(2,5,3)\in a\\ \frac{x-2}{2}=\frac{y-5}{-3}=\frac{z-3}{4}$

2.4

$A(-2,3,4)\in \alpha\\ a:\frac{x-4}{0-4}=\frac{y+2}{2+2}=\frac{z-5}{4-5}\\ \vec{a}=(-4,4,-1)\bot\alpha\\ -4(x+2)+4(y-3)-(z-4)=0\\ -4x+4y-z-16=0$