Further Vectors help (1 Viewer)

Tantrrsh · Dec 21, 2019

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 · Dec 21, 2019

What do you end up with in your working?

InteGrand · Dec 21, 2019

Tantrrsh said:
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

$\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).$

Tantrrsh · Dec 21, 2019

Trebla said:
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 · Dec 21, 2019

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 · Dec 21, 2019

Tantrrsh said:
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 · Dec 21, 2019

InteGrand said:
You should probably show us the steps in your working, in order for us to be able to answer that.

InteGrand · Dec 21, 2019

Tantrrsh said:
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°.

$\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).)$

Further Vectors help (1 Viewer)

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