Related Formula
Componendo and Dividendo rule states that if (x)/(y) = (p)/(q)$\frac{x}{y} = \frac{p}{q}$, then:
(x+y)/(x-y) = (p+q)/(p-q)$$\frac{x+y}{x-y} = \frac{p+q}{p-q}$$
Core Logic
Given expression:
| a + b| + | a - b|| a + b| - | a - b| = √(2) + 11$$\frac{|\vec{a} + \vec{b}| + |\vec{a} - \vec{b}|}{|\vec{a} + \vec{b}| - |\vec{a} - \vec{b}|} = \frac{\sqrt{2} + 1}{1}$$
Applying Componendo and Dividendo:
2| a + b|2| a - b| = (√(2) + 1) + 1(√(2) + 1) - 1 = √(2) + 2√(2) = 1 + √(2)$$\frac{2|\vec{a} + \vec{b}|}{2|\vec{a} - \vec{b}|} = \frac{(\sqrt{2} + 1) + 1}{(\sqrt{2} + 1) - 1} = \frac{\sqrt{2} + 2}{\sqrt{2}} = 1 + \sqrt{2}$$
Squaring both sides:
| a + b|² = (1 + √(2))² | a - b|²$$|\vec{a} + \vec{b}|^2 = (1 + \sqrt{2})^2 |\vec{a} - \vec{b}|^2$$
| a + b|² = (3 + 2√(2)) | a - b|²$$|\vec{a} + \vec{b}|^2 = (3 + 2\sqrt{2}) |\vec{a} - \vec{b}|^2$$
Step 1: Vector Expansion
Expanding using dot products, keeping in mind that | a| = | b|$|\vec{a}| = |\vec{b}|$:
| a|² + | b|² + 2 a· b = (3 + 2√(2))(| a|² + | b|² - 2 a· b)$$|\vec{a}|^2 + |\vec{b}|^2 + 2\vec{a}\cdot\vec{b} = (3 + 2\sqrt{2})(|\vec{a}|^2 + |\vec{b}|^2 - 2\vec{a}\cdot\vec{b})$$
2| a|² + 2 a· b = (3 + 2√(2))(2| a|² - 2 a· b)$$2|\vec{a}|^2 + 2\vec{a}\cdot\vec{b} = (3 + 2\sqrt{2})(2|\vec{a}|^2 - 2\vec{a}\cdot\vec{b})$$
2| a|² (1 - (3 + 2√(2))) = -2 a· b (1 + 3 + 2√(2))$$2|\vec{a}|^2 (1 - (3 + 2\sqrt{2})) = -2\vec{a}\cdot\vec{b} (1 + 3 + 2\sqrt{2})$$
Simplifying directly leads to:
a· b| a|² = 2 + 2√(2)4 + 2√(2) = 1√(2)$$\frac{\vec{a}\cdot\vec{b}}{|\vec{a}|^2} = \frac{2 + 2\sqrt{2}}{4 + 2\sqrt{2}} = \frac{1}{\sqrt{2}}$$
Step 2: Final Calculation
We need to find | a + b|²| a|²$\frac{|\vec{a} + \vec{b}|^2}{|\vec{a}|^2}$:
| a + b|²| a|² = | a|² + | b|² + 2 a· b| a|² = 1 + 1 + 2 a· b| a|²$$\frac{|\vec{a} + \vec{b}|^2}{|\vec{a}|^2} = \frac{|\vec{a}|^2 + |\vec{b}|^2 + 2\vec{a}\cdot\vec{b}}{|\vec{a}|^2} = 1 + 1 + \frac{2\vec{a}\cdot\vec{b}}{|\vec{a}|^2}$$
= 2 + 2( 1√(2)) = 2 + √(2)$$= 2 + 2\left(\frac{1}{\sqrt{2}}\right) = 2 + \sqrt{2}$$
Pattern Recognition
Whenever symmetric sums and differences like | x|+| y|$|\vec{x}|+|\vec{y}|$ and | x|-| y|$|\vec{x}|-|\vec{y}|$ occur in ratios, Componendo-Dividendo should be applied immediately to isolate the ratio of the individual magnitudes.
Chapter Mix
Class 12 Physics: Vector Algebra
Class 12 Mathematics: Vector Algebra
