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 are \(\mathbf{A} \mathbf{u}, \mathbf{A} \mathbf{v}\) in that order? (Your answers should be numerical.)
We know that the length of vector u is |u| = 3 and the length of vector v is |v| = 2. Also, the angle between u and v is 60 degrees. We can use this information to find the dot product of u and v, which will help us find their projections.
The dot product of u and v is given by:
u · v = |u| |v| cos θ
where θ is the angle between u and v. Substituting the values given in the problem, we get:
u · v = 3 × 2 × cos 60° = 3
Now, we can find the projection of u onto v using the formula:
projv(u) = (u · v / |v|) v
Substituting the values we know, we get:
projv(u) = (3/2) v
Similarly, we can find the projection of v onto u using the formula:
proju(v) = (v · u / |u|) u
Substituting the values we know, we get:
proju(v) = (3/2) u
Now, let's write the matrix A using the given rows:
A = [ u v ]
So, Au is the first row of A multiplied by u:
Au = (u · u, u · v) = (9, 6)
And Av is the second row of A multiplied by v:
Av = (u · v, v · v) = (6, 4)
