? -4 ? , and w⃗ = < b m a t r i x >. Let v⃗ =

? -4 ? , and \(\vec{w} = \begin{bmatrix} 4 \\ -4 \\ -5 \end{bmatrix}\). Let \(\vec{v} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}\), \(\vec{u} = \begin{bmatrix} 1 \\ -1 \\ 2 \end{bmatrix}\) Evaluate \(\text{proj}_{\vec{v}} \vec{w} + \text{proj}_{\vec{u}} \vec{w}\). Note: Give an exact answer for each of the boxes below, that means don't write 1.68, write 5/6 instead. ?? ?? ??

Transcript

00:02 Hi here for the given question.

00:04 We are given that you is equal to minus 2 0 4 v is equal to 3 minus 1 6 and w is equal to 2 minus 5 and 5.

00:15 Now here we need to calculate the distance between minus 3 u and v plus 5 w.

00:27 So here in our case in order to calculate the distance between this we have here the value of minus 3 u minus v plus 5 times w which is equal to minus 3 multiplied with minus 2 0 4 minus v which is 3 1 minus 6 3 minus 1 plus 6 minus 5 times 2 minus 5 minus 5.

00:57 So here in our case now on simplifying this here we can say that value of 3 u minus v plus 5 times w equals to minus 7 26 and 7.

01:09 So here the value of the distance v is equal to norm of this value.

01:15 So here we have under root of minus 7 square plus 26 square plus 7 square.

01:22 So this is equal to under root of 774.

01:25 So here the value of this distance v is equal to 3 times under root of 86.

01:33 So this is the first part solution now here further for the next part.

01:38 We need to calculate what is the value of u dot v multiplied with w.

01:51 So first of all, we will calculate the dot product.

01:53 So here the dot product of u and v will be minus 2 multiplied with 3 plus 0 multiplied with minus 1 plus 4 multiplied with 6 and the value is again multiplied with w.

02:04 So here we have 18 times w.

02:06 So this is equal to 18 multiplied with 2 minus 5 and minus 5.

02:11 So here this value is equal to 36 minus 90 and minus 90.

02:15 Now further we need to calculate the value of minus 5 v plus w.

02:20 So this is equal to minus 5 times 3 minus 1 and 6 plus 2 minus 5 and minus 5.

02:28 So here by calculating this value here, we can see that the value of minus 5 v plus w is equal to minus 15 5 comma minus 30 plus 2 comma minus 5 comma minus 5.

02:46 So this is equal to minus 13 0 and 35.

02:54 This is negative 35.

02:56 So here in our case now, we need to take the cross product of this to need to find the cross product of minus 5 v plus w with u dot v multiplied with w.

03:07 So this is equal to cross product of minus 13 0 minus 35 and here we have 36 minus 90 and minus 90.

03:20 So after taking the cross product and simplifying this here, we can see that we have this value equals to minus 3 ,150 comma minus 2 ,430 comma 1170.

03:34 So here this is the solution for the first part.

03:38 We need to calculate what is the value of projection of u from v and v onto u.

03:49 So here for the second part, we are given that the value of u is equal to 0 1 3 minus 6 and v is equal to minus 1 1 2 and 2.

03:59 Now we need to calculate the value of the projection.

04:02 So here in our case the formula to calculate the projection of v on u is equal to u v divided by norm of vector v multiplied with v.

04:12 So here in our case the dot product of u and v is equal to minus 5 and the norm of v square is equal to 10...

Question

Please give Ace some feedback