Vector magnitude subtraction

Question

Please help! This was in a textbook and I cannot seem to make sense of it. Would it not be 2?

The two vectors a and b are perpendicular. If a has magnitude 8 and b has magnitude 3, what is |a−2b|??

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what is the dimension of vector ? how many coordinates it has — dato datuashvili, Apr 08 '17 at 19:10
$\mathbf a-\mathbf b$ forms the third side of a right triangle; $\mathbf a-2\mathbf b$ also forms the hypotenuse of a right triangle. What are the lengths of its other two sides? — amd, Apr 08 '17 at 23:02

score 0 · Answer 1 · answered Apr 08 '17 at 19:16

i will use brute force method let us suppose we have following vector

$a=[{\sqrt{64},0,0}]$

$b=[ 0, \sqrt{9}, 0]$

they are orthogonal $a*b=0$

now $a-2*b=[ \sqrt(64), -2*\sqrt{9}, 0]$

length of this will be $\sqrt{64+4*9+0}$

which is equal to $\sqrt{100}=10$

Logan S. · Answer 2 · 2017-04-09T01:01:35.470

Since your vectors are perpendicular, they don't directly add and subtract like scalars would. Denoting them in R$^2$:

$\textbf{a} - 2\textbf{b} = \begin{pmatrix} 8 \\ 0 \end{pmatrix} - 2 \begin{pmatrix} 0 \\ 3 \end{pmatrix} = \begin{pmatrix} 8 \\ -6 \end{pmatrix}.$

Thus we need to consider what the value of $\begin{pmatrix} 8 \\ -6 \end{pmatrix}$.

Often, this is calculated with the Euclidean Distance, as $\left| \begin{pmatrix} 8 \\ -6 \end{pmatrix} \right| = \sqrt{8^2 + (-6)^2} = 10. $

2 Answers2