JEEOlympiad
Let a = (2, 1, -2) and b = (1, 3, 2). Over all unit vectors c that…
Question
Let a = (2, 1, -2) and b = (1, 3, 2). Over all unit vectors c that are perpendicular to w = (1, 1, 1), the scalar triple product [a b c] attains a maximum value M. Then M^2 = m/n in lowest terms with m, n positive integers. Find m + n.
✓ Verified answer: 329checked by our engine — not a guess
Step-by-step solution
Note [a b c] = c·(a × b) = c·v where v = a × b.
We maximize c·v over unit vectors c with c·w = 0; the maximum is the length of the projection of v onto the plane perpendicular to w, namely M = sqrt(|v|^2 - (v·w)^2/|w|^2), so M^2 = |v|^2 - (v·w)^2/|w|^2.
v = a × b = (1·2 - (-2)·3, (-2)·1 - 2·2, 2·3 - 1·1) = (8, -6, 5).
|v|^2 = 64 + 36 + 25 = 125; v·w = 8 - 6 + 5 = 7; |w|^2 = 3.
M^2 = 125 - 49/3 = (375 - 49)/3 = 326/3.
Then m + n = 326 + 3 = 329.
Final answer329
Stuck on a problem like this?
Paste any JEE or NEET question — verified working, a confidence %, and an honest “not sure” instead of a bluff.
Solve my doubt →More Vectors 3d solutions
Let a = (1, 2, -1), b = (2, -1, 3) and c = (t, 1, 2), where t is a…JEE · MathLet a, b, c be unit vectors with a·b = 1/4, b·c = 1/3 and c·a = 1/5.…JEE · MathConsider the two skew lines L1: r = (1, 2, 3) + s(2, 1, -1) and L2: r…JEE · MathLet a = (3, -1, 2) and b = (1, 5, 1). There is a unique vector r…JEE · MathLet Π be the plane through the three points A = (1, 0, 2), B = (3,…JEE · MathAmong all planes that contain the line L: (x - 1)/2 = (y - 2)/3 = (z…JEE · MathLet a = (1, 1, 0), b = (0, 1, 1), c = (1, 0, 1). The vector d = (2,…JEE · MathLet a = (1, -2, 1), b = (2, 1, -1), c = (1, 1, 2), d = (3, -1, 2).…JEE · Math