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Agniveer Air Force Vector Algebra — practice questions

Explore 41 free Vector Algebra questions in Agniveer Air Force 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 1–30 of 41.

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 →A quantity having only magnitude (no direction) is called:Read the answer and worked solution →Vector addition is:Read the answer and worked solution →The dot product ā · b̄ is:Read the answer and worked solution →The cross product ā × b̄ is:Read the answer and worked solution →What is î × ĵ?Read the answer and worked solution →The magnitude of the vector $3\hat{i} + 4\hat{j}$ is:Read the answer and worked solution →The magnitude of the cross product $|\vec{a}\times\vec{b}|$ equals:Read the answer and worked solution →The unit vector in the direction of $3\hat{i} + 4\hat{j}$ is:Read the answer and worked solution →The dot product of $(2\hat{i}+3\hat{j}+\hat{k})$ and $(\hat{i}-\hat{j}+\hat{k})$ is:Read the answer and worked solution →The value of $\hat{i}\times\hat{j}$ is:Read the answer and worked solution →The magnitude of the vector $\hat{i}+\hat{j}+\hat{k}$ is:Read the answer and worked solution →Find the odd quantity in each rowRead the answer and worked solution →Match each kind of vector to its definitionRead the answer and worked solution →Does this produce a number or a vector?Read the answer and worked solution →True or false: the zero vectorRead the answer and worked solution →Complete the definition of a unit vectorRead the answer and worked solution →Tap every statement true of the scalar productRead the answer and worked solution →The two productsRead the answer and worked solution →Complete the angle formulaRead the answer and worked solution →True or false: scalar multiplicationRead the answer and worked solution →Order the steps: build a unit vector from a pointRead the answer and worked solution →Match each law to what it saysRead the answer and worked solution →Which vectors are these?Read the answer and worked solution →True or false: vector vocabularyRead the answer and worked solution →Spot the one that does not belongRead the answer and worked solution →Assertion and reasonRead the answer and worked solution →Two are true, one is notRead 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. $|\vec{a}| = |\vec{b}|$
Show answer and solution

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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