Question

If vec A = 2i + k and B = i + 3j - 9k then express vec B = V₁ - 2V₂ where vec V₁ is perpendicular to B and V₂ is parallel to vec A.

Answer 1

Given \(\vec A=2\hat i+\hat k\)

\(\vec B\) = \(\hat i+3\hat j-9\hat k\)

\(\vec V_2\) is parallel to \(\vec A\).

\(\therefore\) \(\vec V_2\) = \(\lambda(2\hat i+\hat k)\) 

\(\vec V_1\) is perpendicular to \(\vec B\)

\(\vec V_1.\vec B=0\)----(1)

Let \(\vec V_1\) = x\(\hat i\) + y\(\hat j\) + z\(\hat k\)

Given that \(\vec B=\) V₁ - 2V₂

⇒ (\(\hat i+3\hat j-9\hat k\)) = (x\(\hat i\) + y\(\hat j\) + z\(\hat k\)) - 2\(\lambda\)(2\(\hat i\) + \(\hat k\))

⇒ \(\hat i+3\hat j-9\hat k\) = (x - h\(\lambda\))\(\hat i\) + y\(\hat j\) + (z - 2\(\lambda\))\(\hat k\)

\(\therefore\) x - 4\(\lambda\) = 1, y = 3

z - 2\(\lambda\) = -9------(2)

From (1), \(\vec V_1,\vec B=0\)

\((x\hat i+y\hat j+z\hat k).(\hat i+3\hat j-9\hat k)=0\)

⇒ x + 3y - 9z = 0

⇒ 1 + 4\(\lambda\) + 9 - 9(-9 + 2\(\lambda\)) = 0 (From(2))

⇒ 10 + 4\(\lambda\) + 81 + 18\(\lambda\) = 0

⇒ 14\(\lambda\) = 91

⇒ \(\lambda\) = 91/14 = 13/2

\(\therefore\) x = 1 + 4\(\lambda\) = 1 + 4 x 13/2 = -9 + 13 = 4

\(\therefore\) V₁ = 27\(\hat i\) + 3\(\hat j\) + 4\(\hat k\)

V₂ = 13/2(2\(\hat i\) + \(\hat k\)) = 13\(\hat i\) + 13/2 \(\hat k\)