Help with finding parallel vectors

[Aske]

Homework Statement

The question states "let u= 2i+mj-10k and v=i-3j+nk, find the value of n and m such that u,v are parallel", the second part states asks the same but "u,v are perpendicular"

Homework Equations

The Attempt at a Solution

I attempted to use a dot product solution I guess, because vectors u+v should equal 0 when perpendicular. I'm lost on what to actually do. Thanks in advance.

 

[RoyalCat]

Remember the definition of the dot product:

$$\vec A \cdot \vec B = A_x B_x + A_y B_y + A_z B_z$$

As you correctly stated, for two perpendicular vectors, the dot product is 0.

Let two vectors, $$\vec w, \vec q$$ be parallel. That means that they're both in the same direction, and therefore, they only differ by some scalar factor R (For instance, $$\vec q$$ could be 2 times longer than $$\vec w$$ (R=2) or it could be the same length, but anti-parallel (A negative R value of -1 would achieve that goal) or 2 times longer, but anti-parallel (R=-2)).

So in general, we can write: $$\vec q=R\vec w$$
Note that we've written a vector equation. That's actually 3 scalar equations in one. Simply solve for your three variables, $$R, m, n$$ and you're done.

What's important is that you understand how we've identified parallel\anti-parallel vectors and perpendicular ones. Are these two points clear to you?

 

[ehild]

If two vectors

u=x1 i +y1 j + z1 k and v=x2 i +y2 j + z2 k

are parallel, then one of them is a multiple of the other

u=a*v (a is a scalar).

That means the same for all components:


x1=a*x2
y2=a*y2
z1=a*z2.

The two vectors are perpendicular if their dot product is 0 which means

x1x2 + y1y2+ z1z2 = 0.


ehild

 

[Aske]

Remember the definition of the dot product:

$$\vec A \cdot \vec B = A_x B_x + A_y B_y + A_z B_z$$

As you correctly stated, for two perpendicular vectors, the dot product is 0.

Let two vectors, $$\vec w, \vec q$$ be parallel. That means that they're both in the same direction, and therefore, they only differ by some scalar factor R (For instance, $$\vec q$$ could be 2 times longer than $$\vec w$$ (R=2) or it could be the same length, but anti-parallel (A negative R value of -1 would achieve that goal) or 2 times longer, but anti-parallel (R=-2)).

So in general, we can write: $$\vec q=R\vec w$$
Note that we've written a vector equation. That's actually 3 scalar equations in one. Simply solve for your three variables, $$R, m, n$$ and you're done.

What's important is that you understand how we've identified parallel\anti-parallel vectors and perpendicular ones. Are these two points clear to you?

I somewhat understand what you are saying, so are you saying I need to isolate the unknowns? I still don't fully understand this.

 

[Juan Pablo]

Yes, solve the linear system of equations.

 

[Aske]

Ok, trying this with the perpendicular qustion it'd be "(2i+mj-10k) . (i-3j+nk)=0". I would have 2 unknowns, how would I solve this. I still don't know how to go about the parallel question.