For the vectors a = 1, 4 and b = 2, 3 , find orth ab.

Solution:

Given, vectors a = (1, 4) and b = (2, 3)

We have to find orth ab.

Orth_ab = b - proj_ab

proj_ab = \(\frac{a.b}{\left | a \right |^{2}}*a\)

Now, a.b = (1, 4).(2, 3)

= 1(2) + 4(3)

= 2 + 12

a.b = 14

|a| is the modulus of vector a.

\(\left | a \right |=\sqrt{1^{2}+4^{2}}\)

\(\left | a \right |=\sqrt{1+16}\)

\(\left | a \right |=\sqrt{17}\)

\(\left | a \right |^{2}=(\sqrt{17})^{2}\)

\(\left | a \right |^{2}=17\)

proj_ab = \(\frac{14}{17}.(1, 4)\)

= \(\frac{14\times 1}{17},\frac{14\times 4}{17}\)

= \(\frac{14}{17},\frac{56}{17}\)

proj_ab = \(\frac{14}{17},\frac{56}{17}\)

Orth_ab = (2, 3) - \(\frac{14}{17},\frac{56}{17}\)

\(= 2-\frac{14}{17},3-\frac{56}{17}\)

= \(\frac{20}{17},\frac{-5}{17}\)

Orth_ab = (20/17, -5/17)

Therefore, Orth_ab = (20/17, -5/17)

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For the vectors a = 1, 4 and b = 2, 3 , find orth ab.

Summary:

For the vectors a = (1, 4) and b = (2, 3), orth ab is (20/17, -5/17).