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

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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?

Jul 18, 2022

#1
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$$\mathbf{A} \mathbf{u} = \begin{pmatrix} 6 \\ 9 \end{pmatrix} \quad \mathbf{A} \mathbf{v} = \begin{pmatrix} 8 \\ 4 \end{pmatrix}$$

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Jul 18, 2022
#2
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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, $$=$$  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!

Jul 19, 2022
#3
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I got Av= (9;3)

Guest Jul 20, 2022
#4
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Edit: Sorry I meant Au= 3;9

Guest Jul 20, 2022
#5
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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>

Jul 21, 2022
#6
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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.)

Jul 21, 2022