Lagrange Identity

Question

For two vectors with magnitudes 4 and 3 enclosing a 30 degree angle, evaluate the sum of squares |a . b|^2 plus |a x b|^2.

RankGuru · Accepted Answer

The Lagrange identity states that |a . b|^2 + |a x b|^2 = |a|^2 |b|^2, since the squared cosine and squared sine of the included angle add to one. This elegant relation often shortcuts JEE Advanced problems that appear to demand separate dot and cross computations. By definition |a . b|^2 = |a|^2|b|^2 cos^2(theta) and |a x b|^2 = |a|^2|b|^2 sin^2(theta); adding them and factoring gives |a|^2|b|^2(cos^2 theta + sin^2 theta) = |a|^2|b|^2. Thus the angle becomes irrelevant, and the sum equals (4^2)(3^2) = 16 times 9 = 144. Option 108 is obtained by computing only the cross-product square, 144 times sin... actually 144 times (1/4) plus an error. Option 36 comes from |a|^2 |b|^2 without one factor, using 4 times 9. Option 72 halves the correct product mistakenly. The fact that the 30-degree angle does not appear in the final answer is the central insight the problem rewards. Plausibility check: computing directly, |a . b|^2 = (12 cos 30)^2 = (6 sqrt 3)^2 = 108 and |a x b|^2 = (12 sin 30)^2 = 6^2 = 36, and 108 + 36 = 144, confirming the identity.

Lagrange Identity

Select the correct option:

🔒 Solution Hidden from View

More vector algebra Practice Questions

The shortest distance between two skew lines is computed using which expression involving their dire...

Two vectors of magnitudes 5 and 12 act at a point with an included angle of 90 degrees; evaluate the...

If the scalar triple product [a b c] equals 7 for three given vectors, what is the value of the rear...

Three unit vectors a, b and c satisfy a + b + c = 0; using this constraint, evaluate the sum a . b +...

Three vectors a = i + 2j + 3k, b = 2i + j + k and c = 3i + j + 2k are given; compute their scalar tr...

If two nonzero vectors a and b satisfy |a + b| = |a - b|, what can be concluded about the angle betw...

Consider the vectors a = 3i + 4j and b = i + 2j + 2k; find the scalar projection of vector a onto th...

What is the unit vector directed along the vector obtained by adding a = 2i - j + 2k and b = i + j -...

For what value of the scalar lambda are the three vectors i + j + k, i + lambda j + 2k and 2i + j + ...

The position vectors of three points are A(1,1,1), B(2,3,5) and C(3,5,1); find the area of triangle ...

Using the standard expansion identity, simplify the vector triple product a x (b x c) where a, b and...

A point P divides the line segment joining A(1,2,3) and B(4,8,9) internally in the ratio 2:1; locate...

About This Question

Solution