How to solve exercises with 2D and 3D vectors and dot product

[Kalbaan]

Hey!

I just joined the forum, but would like to get some help with 2D&3D vectors and dot product. I missed some classes due to a bad illness and now can't get the hang of it at all..
Would appreciate it a lot, if someone could explain me how to solve these 5 exercises.
View attachment 1783

 

[Deveno]

Some hints:

a) Can you think of a way to use the dot product here?

b) If two vectors are of equal magnitude, but opposite direction, their vector sum is 0. Why is this relevant?

c) We have the 3 equations:

$3\lambda + 3\mu + 1 = x$

$-\lambda + 2\mu - 1 = y$

$4\lambda - 2 = z$

By multiplying equations (1) and (2) by suitable integers, can you eliminate $\mu$? Then try to use that equation and equation 3 to eliminate $\lambda$.

d) Such a line should be parallel to $v$, right?

e) Think about what the direction vectors of such a plane have to be...

 

[Kalbaan]

Thanks a lot! You made my week mate!
Got the hang of them with your hints and my teachers powerpoint shows.

 

[Evgeny.Makarov]

There is another way of finding an equation of a plane in (c) and (e): an equation of the plane perpendicular to $(A,B,C)$ and passing through $(x_0.y_0,z_0)$ is $A(x-x_0)+B(y-y_0)+C(z-z_0)=0$, or $Ax+By+Cz+(-Ax_0-By_0-Cz_0)=0$.