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JEE Advanced Vector Algebra — practice questions

Explore 4 free Vector Algebra questions in JEE Advanced Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Vector Algebra questions and exercises with answers

Showing questions 1–4 of 4.

The dot product (scalar product) of two vectors $\vec{a}$ and $\vec{b}$ is:Read the answer and worked solution →Two non-zero vectors $\vec{a}$ and $\vec{b}$ are perpendicular if and only if:Read the answer and worked solution →The cross product $\vec{a} \times \vec{b}$ of two vectors has magnitude:Read the answer and worked solution →Two vectors of magnitudes $3$ and $4$ are perpendicular to each other. The magnitude of their resultant is:Read the answer and worked solution →

Worked practice questions

The dot product (scalar product) of two vectors $\vec{a}$ and $\vec{b}$ is:

  1. $|\vec{a}||\vec{b}|\cos\theta$
  2. $|\vec{a}|^2 + |\vec{b}|^2$
  3. $|\vec{a}||\vec{b}|\sin\theta$
  4. $|\vec{a}||\vec{b}|\tan\theta$
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Correct answer: A. $|\vec{a}||\vec{b}|\cos\theta$

The dot product is defined as $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$ where $\theta$ is the angle between them. The result is a scalar. The sine version (option A) defines the magnitude of the cross product, which is a vector, not a scalar. Mixing up the two is a recurring exam trap.

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Two non-zero vectors $\vec{a}$ and $\vec{b}$ are perpendicular if and only if:

  1. $\vec{a} \times \vec{b} = 0$
  2. $\vec{a} \cdot \vec{b} = 0$
  3. $\vec{a} + \vec{b} = 0$
  4. $|\vec{a}| = |\vec{b}|$
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Correct answer: B. $\vec{a} \cdot \vec{b} = 0$

$\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$. For two non-zero vectors, this is zero exactly when $\cos\theta = 0$, i.e. $\theta = 90°$. Option B describes *parallel* vectors (cross product zero). Option D would require the vectors to be exact opposites of each other, not just perpendicular.

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The cross product $\vec{a} \times \vec{b}$ of two vectors has magnitude:

  1. $|\vec{a}||\vec{b}|\cos\theta$
  2. $|\vec{a}||\vec{b}|\sin\theta$
  3. $|\vec{a}| + |\vec{b}|$
  4. $|\vec{a}|^2 - |\vec{b}|^2$
Show answer and solution

Correct answer: B. $|\vec{a}||\vec{b}|\sin\theta$

$|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta$. The cross product is a vector perpendicular to both $\vec{a}$ and $\vec{b}$, with direction given by the right-hand rule. Its magnitude equals the area of the parallelogram spanned by $\vec{a}$ and $\vec{b}$. Geometric reading: when $\theta = 0$ or $\pi$ (parallel or anti-parallel), $\sin\theta = 0$ and the cross product vanishes, which is why parallel vectors have zero cross product.

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More Mathematics topics

Vector Algebra and 3D GeometryStatistics and ProbabilityContinuity, Differentiability & DerivativesIntegrals and ApplicationsMatrices and DeterminantsRelations, Functions & Inverse TrigonometryStraight Lines and Conic SectionsDeterminants