Vectors (JEE Advanced)

Question 1

Let a⃗ and b⃗ be vectors with lvert a⃗rvert =2, lvert b⃗rvert =3 and a⃗·b⃗=3. A vector r⃗ satisfies r⃗×a⃗=b⃗×a⃗ and r⃗·b⃗=0. The value of lvert r⃗rvert is ______. Exactly one option is correct.

Accepted Answer

C. r⃗×a⃗=b⃗×a⃗ gives (r⃗−b⃗)×a⃗=0⃗. Since a⃗≠0⃗, this says only that r⃗−b⃗ is a scalar multiple of a⃗ (possibly 0⃗): r⃗=b⃗+ta⃗ for some real t. It does not give r⃗=b⃗.
Then r⃗·b⃗=lvert b⃗rvert ²+t a⃗·b⃗=9+3t=0, so t=−3 and r⃗=b⃗−3a⃗.
lvert r⃗rvert ²=lvert b⃗rvert ²−6 a⃗·b⃗+9lvert a⃗rvert ²=9−18+36=27, so lvert r⃗rvert =3√3. Answer (C).
Difficulty: Hard

Question 2

Let λ be a non-zero real number and let a⃗,b⃗ be arbitrary vectors. For every such choice, the equation λx⃗+a⃗×x⃗=b⃗ has exactly one solution x⃗, and this solution is ______. Exactly one option is cor

Accepted Answer

B. Dot the equation with a⃗: λ(a⃗·x⃗)=a⃗·b⃗, so a⃗·x⃗=(a⃗·b⃗)/λ.
Cross a⃗ with the equation: λ(a⃗×x⃗)+a⃗×(a⃗×x⃗)=a⃗×b⃗, i.e. λ(a⃗×x⃗)+(a⃗·x⃗)a⃗−lvert a⃗rvert ²x⃗=a⃗×b⃗.
Substitute a⃗×x⃗=b⃗−λx⃗: λb⃗−λ²x⃗+(a⃗·b⃗)/λa⃗−lvert a⃗rvert ²x⃗=a⃗×b⃗, so
(λ²+lvert a⃗rvert ²)x⃗=λb⃗+(a⃗·b⃗)/λa⃗−a⃗×b⃗, x⃗=(λ²b⃗+(a⃗·b⃗)a⃗−λ(a⃗×b⃗))/(λ(λ²+lvert a⃗rvert ²)).
Since λ²+lvert a⃗rvert ²>0 every step is reversible, so this

Question 3

Let b⃗ and c⃗ be non-parallel vectors, let n⃗=b⃗×c⃗, and let r⃗ be any vector. For every such choice, the orthogonal projection of r⃗ onto the plane through the origin containing b⃗ and c⃗ is ______.

Accepted Answer

A. n⃗≠0⃗ is normal to the plane, so the projection is r⃗−((r⃗·n⃗))/(lvert n⃗rvert ²)n⃗.
Using (u⃗×v⃗)×w⃗=(u⃗·w⃗)v⃗−(v⃗·w⃗)u⃗: (n⃗×r⃗)×n⃗=(n⃗·n⃗)r⃗−(r⃗·n⃗)n⃗. Dividing by lvert n⃗rvert ² gives exactly the projection. Answer (A).
(B) is the negative of (A). (C) is the component along the normal. (D) is correct only when b⃗⊥c⃗; e.g. b⃗=î, c⃗=î+ĵ, r⃗=î gives î+1/2(î+ĵ)≠î.
Difficulty: Hard

Vectors: questions with solutions

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