Brennan Flores

2021-01-27

If (v-ku) is orthogonal to v, then what is k??

wheezym

Step 1
We have are two column matrices .
We have to find the value of k so that (v-ku) is orthogonal to v.
Step 2
First observe that we are given two column matrices u and v. We know that column matrices simply represents a vector in the said space(here ${\mathbb{R}}^{3}$) .So here u and v are vector in ${\mathbb{R}}^{3}$.
Now we know that two vectors are orthogonal if their dot product is 0.
Now (v-ku) gives the vector $\left[\begin{array}{c}2-k\\ -1-k\\ 2-k\end{array}\right]$ and according to the given problem ,this is orthogonal to
$v=\left[\begin{array}{c}2\\ -1\\ 2\end{array}\right]$
So, $\left[\begin{array}{c}2-k\\ -1-k\\ 2-k\end{array}\right]\left[\begin{array}{c}2\\ -1\\ 2\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$
Since this is simply a vector dot product ,so we can write :
$2×\left(2-k\right)-1×\left(-1-k\right)+2×\left(2-k\right)=0$
$4-2k+1+k+4-2k=0$
$-3k+9=0$
$-3k=-9$
$3k=9$
$k=3$
Which is required value of k and for this value of k (v-ku) is orthogonal to v.

Jeffrey Jordon

Answer is given below (on video)

Jeffrey Jordon

Answer is given below (on video)

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