I really need help with this question, I do not understand matrices.

Let \(\mathbf{u}\) and \(\mathbf{v}\) be vectors such that \(\|\mathbf{u}\| = 3\) and \(\|\mathbf{v}\| = 2\) such that the angle between \(\mathbf{u} \) and \(\mathbf{v}\) when placed tail to tail is \(60^\circ\).

Let \(\mathbf{A}\) be a matrix such that

\(\mathbf{row}_1(\mathbf{A}) = \mathbf{u}, \mathbf{row}_2(\mathbf{A}) = \mathbf{v}.\)

Then what is \(\mathbf{A} \mathbf{u}, \mathbf{A} \mathbf{v}\) in that order?

Thanks in advance!

Guest Jul 18, 2022

#1**-1 **

\(\mathbf{A} \mathbf{u} = \begin{pmatrix} 6 \\ 9 \end{pmatrix} \quad \mathbf{A} \mathbf{v} = \begin{pmatrix} 8 \\ 4 \end{pmatrix}\)

.Guest Jul 18, 2022

#2**+1 **

Hi Guest!

Ok, this question is about matrix multiplicaton and the dot product (Also called: Inner product or scalar product).

**u** is a vector, so it has components: \(u_1,u_2,...u_n\)

Similarly, **v** is a vector, so it has components: \(v_1,v_2,...,v_n\)

Now, no need to actually make it in n-dimension. I mean we can, but why not simplify this and assume we are in 2 dimensional space?

That is, n=2

So, **u \(= \) and v \(= \) **

So our matrix A is: \(A=\begin{bmatrix} u_1 && u_2 \\ v_1 && v_2 \end{bmatrix}\) (As given, u is the first row and v is the second row)

But we are given an angle and lengths of these vectors. Can we find the "dot product"?

Yes: \(u \dot {} v=u_1v_1+u_2v_2=cos(\theta)*\left || u |\right |*\left || v| \right |=cos(60)*3*2=3\)

So, we got: \(u_1v_1+u_2v_2=3\)

Next, let's see what the question really wants:

\(Au=\begin{bmatrix} u_1 && u_2 \\ v_1 && v_2 \end{bmatrix}\) \(*\begin{bmatrix} u_1 \\ u_2 \end{bmatrix}\) = \(\begin{pmatrix} u_1^2+u_2^2\\ u_1v_1+u_2v_2 \end{pmatrix}\) = \(\begin{pmatrix} 3^2\\ 3 \end{pmatrix}=\begin{pmatrix} 9\\ 3 \end{pmatrix}\)

Now, in a similar way, find \(Av\).

Hope this helps!

Guest Jul 19, 2022

#5**+3 **

The dot product of u and v is equal to ||u||* ||v||*cos 60 = 3. (Prove this before you start using it!)

row_1(Au) = \(\mathbf{u \cdot u}\) = 9

row_2(Au) = Dot product of u and v, equaling 3

Au = <9, 3>

Similarly:

row_1(Av) = 3

row_2(Av) = v \cdot v = 4

Av = <3, 4>

Voldemort Jul 21, 2022

#6**+1 **

There seems to be a notational or transcription error in the question.

The statement that the first row of the matrix A is the vector u implies that u is a row vector.

(Similarly, the implication is that v is a row vector.)

We are not given the dimension of u and v, but assuming that it is two, A will be a 2 by 2 matrix.

In that case, the products Au and Av do not exist, for the products to exist u and v would need to be column vectors.

The question would make sense if we were given that

\(\displaystyle \text{row}_{1}(A)=u^{\text{T}},\)

(and similarly for v),

or, calculate

\(\displaystyle \text{A}u^{\text{T}},\)

(If u is a row vector.)

Guest Jul 21, 2022