# Further Vectors help (1 Viewer)

#### Tantrrsh

##### Member
The angle between two vectors u=i-3j and v=ai+5j is 120°. Find a correct to 1 decimal place.
I've tried substituting into the scalar equation but I can't get the answer

#### Trebla

What do you end up with in your working?

#### InteGrand

##### Well-Known Member
The angle between two vectors u=i-3j and v=ai+5j is 120°. Find a correct to 1 decimal place.
I've tried substituting into the scalar equation but I can't get the answer
$\bg_white \noindent \textbf{Hint:} Recall that we have (\mathbf{u}\cdot\mathbf{v})^2 = \left\| \mathbf{u}\right\|^2\left\| \mathbf{v}\right\|^2\cos^2 \theta, where \theta is the angle between the two vectors. You should get a quadratic equation in a from this, which you should know how to solve. Just make sure in general to check which a correctly gives \cos(120^\circ)=\frac{\mathbf{u}\cdot\mathbf{v}}{\left\| \mathbf{u}\right\|\left\| \mathbf{v}\right\|} (although by a rough sketch in this case, you can see there will be two solutions).$

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#### Tantrrsh

##### Member
What do you end up with in your working?
I worked it out, and my answers are 24.4 and -4.4. However, the text book says 4.4 or -24.4. Why is that?

#### Tantrrsh

##### Member
I'm also stuck on this one:
Find vector b given a = 3i - j, a•b = -6 , and the angle between a and b is 30°.

#### InteGrand

##### Well-Known Member
I worked it out, and my answers are 24.4 and -4.4. However, the text book says 4.4 or -24.4. Why is that?
You should probably show us the steps in your working, in order for us to be able to answer that.

#### Tantrrsh

##### Member
You should probably show us the steps in your working, in order for us to be able to answer that.

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#### InteGrand

##### Well-Known Member
I'm also stuck on this one:
Find vector b given a = 3i - j, a•b = -6 , and the angle between a and b is 30°.
$\bg_white \noindent \textbf{Hints:} From the dot product, you should be able to find \left\|\mathbf{b}\right\| (remember the formula \boxed{\mathbf{a}\cdot\mathbf{b}=\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|\cos\theta}). You also know / can work out the angle \mathbf{a} makes to the positive x-axis. Since the angle between \mathbf{a} and \mathbf{b} is known, this gives us two possible values for the angle that \mathbf{b} makes with the positive x-axis, which together with \left\|\mathbf{b}\right\|, will let us work out two possible answers for \mathbf{b}. (Remember, if we know a 2D vector's length and the angle it makes with the positive x-axis in the x-y plane, we can figure out its coordinates using trigonometric formulas, and vice versa (if you're unsure how to do this, it would be best to learn it before tackling this problem).)$

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