# Thread: Vector Calculations

1. ## Vector Calculations

Hello all, not sure if this is the right thread but i need to check some of my answers to some problems. thanks.

if u = i + 2j - k and v = j - 3k are vectors in R^3. Find:

a) a unit vector poiting in the direction of -3u:

my solution: -3 (i + 2j - k) = -3i - 6j +3k

so then the magnitude is: root (9 +36 + 9) which is root (54) so then the unit vector is 1 / root(54) * (3i + 6j - 3k)

b) u * (12v)

i used the fact that u * (12v) is equal to 12 (u * v)
i first found u * v using the dot product formula to get 5. so then 12 (5) = 70.

c) the cosine of the angle between u and v.

I used the product i found earlier (5) and used the formula cos theta = u * v / (absolute value of u * absolute value of v) and got the answer 5 / root (60)

2. ## Re: Vector Calculations

Originally Posted by andrew2322
Hello all, not sure if this is the right thread but i need to check some of my answers to some problems. thanks.

if u = i + 2j - k and v = j - 3k are vectors in R^3. Find:

a) a unit vector poiting in the direction of -3u:

my solution: -3 (i + 2j - k) = -3i - 6j +3k

so then the magnitude is: root (9 +36 + 9) which is root (54) so then the unit vector is 1 / root(54) * (3i + 6j - 3k)

b) u * (12v)

i used the fact that u * (12v) is equal to 12 (u * v)
i first found u * v using the dot product formula to get 5. so then 12 (5) = 70.

c) the cosine of the angle between u and v.

I used the product i found earlier (5) and used the formula cos theta = u * v / (absolute value of u * absolute value of v) and got the answer 5 / root (60)
a) is correct, but you can simplify it further.

b) Is this a dot product or a cross product? Assuming it's a dot product (you should use a dot instead of an asterix), you are correct.

c) You are correct that $\displaystyle \displaystyle \cos{\theta} = \frac{5}{\sqrt{60}}$ which you can simplify more. What is $\displaystyle \displaystyle \theta$?

3. ## Re: Vector Calculations

Hello Prove it.

Yes, B is the dot product, i apologise for the ambiguity.

and with c, if the question asks for the cosine of the angle, does that mean they want cos theta or just theta alone?

4. ## Re: Vector Calculations

Originally Posted by andrew2322
Hello all, not sure if this is the right thread but i need to check some of my answers to some problems. thanks. if u = i + 2j - k and v = j - 3k are vectors in R^3. Find:
a) a unit vector poiting in the direction of -3u:
my solution: -3 (i + 2j - k) = -3i - 6j +3k
so then the magnitude is: root (9 +36 + 9) which is root (54) so then the unit vector is 1 / root(54) * (3i + 6j - 3k)
I actually think that your answer there is not be correct.
Now mind you, it may be just the way I am reading the question.
"a) a unit vector pointing in the direction of -3u"
The word 'pointing in the direction as $\displaystyle -3u$ means in the opposite direction as $\displaystyle u$. That is the effect of the negative.
Positive multiples of $\displaystyle u$ are in the same direction, negative multiples are in the opposite direction
You answer above is a positive multiple.

If $\displaystyle u=i+2j-k$ then $\displaystyle \frac{-1}{\sqrt6}u$ is a unit vector parallel to $\displaystyle -3u$ in the same direction.
That is all part a) is asking for. You did a lot of unnecessary steps.

5. ## Re: Vector Calculations

I understand what you mean Plato, but is it possible that given the information we have, that both solutions are correct in their own right?

6. ## Re: Vector Calculations

Sorry, i think there was a typo in my answer. It was meant to be [-1 / 3 * root(6)] * (3i + 6j - 3k)
which is a unit vector pointing in the right direction as -u.

is this correct?

7. ## Re: Vector Calculations

Originally Posted by andrew2322
is it possible that given the information we have, that both solutions are correct in their own right?
I don't know how to answer that? Consult your text material and/or instructor.

I will say that these are typical questions.
If the question were "write a unit vector in the direction of $\displaystyle -3\vec{u}$" then the simple and correct answer would be $\displaystyle -\frac{\vec{u}}{\|\vec{u}\|}$.

8. ## Re: Vector Calculations

which is equivalent to - (1/3) * (1/ root 6) ( 3i + 6j - 3k) which is the same as (1 / root 6) (i + 2j - k)

is the math correct?