Length of a Vector and Division Formulas

Length of a Vector and Division Formulas

Length of a vector in term of in components

If r = xi + yj + zk then |r| = \(\sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}\).

Proof: Let \(r=\overrightarrow{OP}\) then P = (x, y, z)

Now |r| = OP = \(\sqrt{{{(x-0)}^{2}}+{{(y-0)}^{2}}+{{(z-0)}^{2}}}\),

OP = \(\sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}\).

Examples: r = 2i + 3j – 6k then find |r|

Solution: Given that,

r = 2i + 3j – 6k

Now |r| = \(\sqrt{{{(2-0)}^{2}}+{{(3-0)}^{2}}+{{(6-0)}^{2}}}\),

               = \(\sqrt{{{(2)}^{2}}+{{(3)}^{2}}+{{(6)}^{2}}}\),

               = \(\sqrt{4+9+36}\),

               = \(\sqrt{49}\),

               = 7

Division formulas

• Let a, b be the position vectors of the point A, B respectively. The position vectors of the point P which divided \(\overline{AB}\) in the ratio m : n is  \(\frac{mb+na}{m+n}\). Conversely the point P with the position vector \(\frac{mb+na}{m+n}\) lies on the \(\overline{AB}\) and divides \(\overline{AB}\) in the ratio m : n.
• Let a, b be the position vectors of the point A, B respectively. The position vector of the point P which divides \(\overline{AB}\) externally in the ratio m : n in \(\frac{mb-na}{m-n}\).
• The point which divides the line segment joining (x₁, y₁, z₁) and (x₂, y₂, k₂) ration m : n is internally is \(\left( \frac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\frac{m{{y}_{2}}+n{{y}_{1}}}{m+n},\frac{m{{z}_{2}}+n{{z}_{1}}}{m+n} \right)\).
• The point which divides the line segment joining (x₁ , y₁ , z₁) and (x₂, y₂, k₂) ration m : n externally is \(\left( \frac{m{{x}_{2}}-n{{x}_{1}}}{m-n},\frac{m{{y}_{2}}-n{{y}_{1}}}{m-n},\frac{m{{z}_{2}}-n{{z}_{1}}}{m-n} \right)\).
• The ratio in which xi + yj + zk divides the line segment joining x₁i + y₁j + z₁k and x₂i + y₂j + z₂k is x – x₁ : x₂ – x  = y – y₁ : y₂ – y  = z  – z₁  :  z₂ – z.

Example: Find the position vector of the point which divided the line joining the point 3a – 2b and a + b in the ratio 2 : 1

1. Internally and

2. Externally

Solution: Given that the line joining the point 3a – 2b and a + b in the ratio 2 : 1

We know that   internally divided in ratio \(\frac{mb+na}{m+n}\),

Externally divided in ratio \(\frac{mb-na}{m-n}\),

Position vector the point which divides the line joining the given point in the line ratio 2 : 1 Internally is

= \(\frac{2(a+b)+1(3a-2b)}{2+1}\),

= \(\frac{5a}{3}\),

Position vector the point which divides the line joining the given point in the line ratio 2 : 1 Externally  is

= \(\frac{2(a+b)-1(3a-2b)}{2-1}\),

= – a + 4b.