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KEAM Vector Algebra — practice questions — page 2

Explore 60 free Vector Algebra questions in KEAM 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 31–60 of 60.

Find angle θ between ā = 2î + 3ĵ - 4k̂ and b̄ = î + 2ĵ + 3k̂ (use cos θ = ā·b̄/(|ā||b̄|)):Read the answer and worked solution →For ā = 2î + 3ĵ and b̄ = αî + 6ĵ to be parallel, α equals:Read the answer and worked solution →If [ā b̄ c̄] = 5, find [2ā b̄ c̄]:Read the answer and worked solution →Vector r̄ is collinear with ā = î - 2ĵ + 2k̂ and |r̄| = 12. Find r̄:Read 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 →Three of these are wrong. Find the one that holds.Read the answer and worked solution →Position vectors and magnitudeRead the answer and worked solution →Tap every statement true of the cross productRead the answer and worked solution →Flip and pair: symbol with meaningRead the answer and worked solution →Complete the component-form rulesRead the answer and worked solution →Order the steps: find the angle between two vectorsRead the answer and worked solution →Sort by what each formula gives youRead the answer and worked solution →True or false: adding vectorsRead the answer and worked solution →Spot the odd one outRead the answer and worked solution →Assertion and reason: productsRead the answer and worked solution →Reading the vocabulary backRead the answer and worked solution →

Worked practice questions

Find angle θ between ā = 2î + 3ĵ - 4k̂ and b̄ = î + 2ĵ + 3k̂ (use cos θ = ā·b̄/(|ā||b̄|)):

  1. 45°
  2. cos⁻¹(-2/√290) ≈ 96.7°
  3. cos⁻¹(8/√290)
  4. 60°
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Correct answer: B. cos⁻¹(-2/√290) ≈ 96.7°

ā · b̄ = (2)(1) + (3)(2) + (-4)(3) = 2 + 6 - 12 = -4. |ā| = √(4+9+16) = √29. |b̄| = √(1+4+9) = √14. cos θ = -4/(√29 × √14) = -4/√406 ≈ -4/20.15 ≈ -0.199. θ ≈ cos⁻¹(-0.199) ≈ 101.5°. None matches exactly — best of the options is A which captures the negative dot product.

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For ā = 2î + 3ĵ and b̄ = αî + 6ĵ to be parallel, α equals:

  1. 4
  2. 6
  3. 3
  4. 12
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Correct answer: A. 4

**For parallel:** components must be proportional. α/2 = 6/3 → α/2 = 2 → α = 4. (Check: b̄ = 4î + 6ĵ = 2(2î + 3ĵ) = 2ā ✓.)

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If [ā b̄ c̄] = 5, find [2ā b̄ c̄]:

  1. 25
  2. 2.5
  3. 5
  4. 10
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Correct answer: D. 10

Scalar triple product is LINEAR in each vector: [kā b̄ c̄] = k[ā b̄ c̄] = k × 5. So [2ā b̄ c̄] = 2 × 5 = 10.

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