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!
\(\mathbf{A} \mathbf{u} = \begin{pmatrix} 6 \\ 9 \end{pmatrix} \quad \mathbf{A} \mathbf{v} = \begin{pmatrix} 8 \\ 4 \end{pmatrix}\)
.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!
+289 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>
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.)
