1 mark, −0.333MathematicsVector spaces, rank and eigenvaluesNorms, inner products and orthogonalityModerateabout 2 min

IIT JAM 2021 Question 10: Vector spaces, rank and eigenvalues

Let $\|\cdot\|$ and $\langle\cdot,\cdot\rangle$ denote the standard norm and inner product in $\mathbb{R}^n$ . If $u, v \in \mathbb{R}^3$ with $\|u\| = \|v\| = 2$ and the angle between $u$ and $v$ is $\pi/3$ , then

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Answer and worked solution

Answer: (D)

The inner product is $\langle u, v \rangle = \|u\|\|v\|\cos(\pi/3) = 2\cdot2\cdot\tfrac12 = 2$ .

Then $\|u - v\|^2 = 8 - 2\langle u,v\rangle = 8 - 4 = 4$ , so $\|u-v\| = 2$ , and $\|u + v\|^2 = 8 + 4 = 12$ , so $\|u+v\| = 2\sqrt{3}$ . Only option (D) matches.

The trap: (C)

\|u - v\| = 2\sqrt{3}

$\|u-v\|$ works out to $2$ , not $2\sqrt{3}$ ; the $2\sqrt{3}$ value belongs to $\|u+v\|$ , so (C) attaches the right number to the wrong expression.

Plain-text version

Let ‖·‖ and ⟨·,·⟩ denote the standard norm and inner product in ℝⁿ. If u, v ∈ ℝ³ with ‖u‖ = ‖v‖ = 2 and the angle between u and v is π/3, then (A) ‖u - v‖ = 2√2 (B) ⟨ u, v ⟩ = 2√3 (C) ‖u - v‖ = 2√3 (D) ‖u + v‖ = 2√3

Also written as: Let | * | and <*,*> denote the standard norm and inner product in R^n. If u, v in R^3 with | u | = | v | = 2 and the angle between u and v is pi/3, then

Vector spaces, rank and eigenvalues has come up 41 times in 6 exams since 2008.

What this question tests

This question is on Norms, inner products and orthogonality, part of Vector spaces, rank and eigenvalues in Mathematics.

Length and angle in vector spaces and the matrices that preserve them.

TestStop has 7 past-paper questions on this subtopic.

Take this paper as a timed mock More on Norms, inner products and orthogonality

Answer and worked solution

Answer: (D)

The inner product is $\langle u, v \rangle = \|u\|\|v\|\cos(\pi/3) = 2\cdot2\cdot\tfrac12 = 2$ .

Then $\|u - v\|^2 = 8 - 2\langle u,v\rangle = 8 - 4 = 4$ , so $\|u-v\| = 2$ , and $\|u + v\|^2 = 8 + 4 = 12$ , so $\|u+v\| = 2\sqrt{3}$ . Only option (D) matches.

The trap: (C)

\|u - v\| = 2\sqrt{3}

$\|u-v\|$ works out to $2$ , not $2\sqrt{3}$ ; the $2\sqrt{3}$ value belongs to $\|u+v\|$ , so (C) attaches the right number to the wrong expression.

Plain-text version

Also written as: Let | * | and <*,*> denote the standard norm and inner product in R^n. If u, v in R^3 with | u | = | v | = 2 and the angle between u and v is pi/3, then

Vector spaces, rank and eigenvalues has come up 41 times in 6 exams since 2008.

What this question tests

This question is on Norms, inner products and orthogonality, part of Vector spaces, rank and eigenvalues in Mathematics.

Length and angle in vector spaces and the matrices that preserve them.

TestStop has 7 past-paper questions on this subtopic.

Question from the IIT JAM 2021 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.

IIT JAM 2021 Question 10: Vector spaces, rank and eigenvalues

Answer and worked solution

What this question tests

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IIT JAM 2021 Question 10: Vector spaces, rank and eigenvalues

Answer and worked solution

What this question tests

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