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ISC Class 12 Vector Algebra and 3D Geometry — practice questions — page 4

Explore 120 free Vector Algebra and 3D Geometry questions in ISC Class 12 Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Vector Algebra and 3D Geometry questions and exercises with answers

Showing questions 91–120 of 120.

The angle θ between two PLANES with normals n1 and n2 satisfies:Read the answer and worked solution →The vector triple product a × (b × c) simplifies via:Read the answer and worked solution →The Cartesian equation of a plane with intercepts 2, 3, 6 on the x, y, z axes is:Read the answer and worked solution →For direction cosines l, m, n of any line in 3D:Read the answer and worked solution →The direction cosines of the line joining (1, 2, 3) and (4, 5, 6) areRead the answer and worked solution →The normalising relationRead the answer and worked solution →The scalar triple product [a b c] geometrically represents:Read the answer and worked solution →Scalar or vector?Read the answer and worked solution →The distance of the point (2, 1, -1) from the plane 2x - y + 2z + 5 = 0 isRead the answer and worked solution →The angle between two planes with normals n₁ and n₂ is given byRead the answer and worked solution →Three of these are wrong. Find the one that holds.Read the answer and worked solution →The additive identity for vectors is the:Read the answer and worked solution →Multiplying a vector by a scalar changes its:Read the answer and worked solution →Flip and pair: symbol with meaningRead the answer and worked solution →The perpendicular distance from a point P with position vector p to a line r = a + λb isRead the answer and worked solution →The plane through the points (1,1,0), (1,0,1) and (0,1,1) has equationRead the answer and worked solution →The line through the origin with direction ratios (1, 1, 1) meets the plane x + y + z = 6 atRead the answer and worked solution →The image of the point (1, 2, 3) in the plane x + y + z = 0 isRead the answer and worked solution →Which quantity is which?Read the answer and worked solution →Complete the definition of skew linesRead the answer and worked solution →Tap each statement that is true of a line's equationRead the answer and worked solution →The angle between the planes 2x - y + 2z = 5 and x + 2y - 2z = 4 satisfiesRead the answer and worked solution →Skew linesRead the answer and worked solution →Tap every pair of lines that could be skewRead the answer and worked solution →Which of these is a scalar quantity?Read the answer and worked solution →A UNIT VECTOR in the direction of v = 3i + 4j is:Read the answer and worked solution →True or false: vectors and 3DRead the answer and worked solution →Vector numbersRead the answer and worked solution →Three of these are wrong. Find the one that holds.Read the answer and worked solution →Spot the odd one outRead the answer and worked solution →

Worked practice questions

The angle θ between two PLANES with normals n1 and n2 satisfies:

  1. $\cos \theta = (n_1 · n_2)/(|n_1||n_2|)$
  2. $\sin \theta = (n_1 · n_2)/(|n_1||n_2|)$
  3. $\tan \theta = |n_1 × n_2|/|n_1 · n_2|$
  4. $\theta = n_1 + n_2$
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Correct answer: A. $\cos \theta = (n_1 · n_2)/(|n_1||n_2|)$

Angle between two planes = angle between their normals: cos θ = (n1 · n2)/(|n1||n2|). For line-plane angle, it's sin θ (complement). For two lines, it's cos θ between direction vectors.

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The vector triple product a × (b × c) simplifies via:

  1. $a · (b × c)$ (the scalar triple)
  2. $(a × b) × c$ (always associative)
  3. $(a · c) b − (a · b) c$ (BAC − CAB)
  4. $|a||b||c| \sin \theta$ (single trig)
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Correct answer: C. $(a · c) b − (a · b) c$ (BAC − CAB)

BAC − CAB rule: a × (b × c) = (a · c) b − (a · b) c. The result lies in the plane of b and c. Cross product is NOT associative.

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The Cartesian equation of a plane with intercepts 2, 3, 6 on the x, y, z axes is:

  1. $x/2 + y/3 + z/6 = 1$
  2. $2x + 3y + 6z = 1$
  3. $x + y + z = 11$
  4. $x/3 + y/2 + z/6 = 1$
Show answer and solution

Correct answer: A. $x/2 + y/3 + z/6 = 1$

**Intercept form of plane:** x/p + y/q + z/r = 1 where p, q, r are intercepts on x, y, z axes. **So x/2 + y/3 + z/6 = 1.**

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