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

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 61–90 of 120.

The vector equation of a line passing through point a and parallel to vector b isRead the answer and worked solution →The equation of a plane through the intersection of π₁ = 0 and π₂ = 0 has the formRead the answer and worked solution →The line r = (2, -1, 4) + λ(3, 0, 2), expressed in symmetric Cartesian form, isRead the answer and worked solution →Spot the one that does not belongRead the answer and worked solution →The Cartesian equation (x-1)/2 = (y-2)/3 = (z-3)/4 representsRead the answer and worked solution →The angle θ between two lines with direction vectors b₁ and b₂ is given byRead the answer and worked solution →The distance of the point (x₀, y₀, z₀) from the plane ax + by + cz + d = 0 isRead the answer and worked solution →The shortest distance between skew lines r = a₁ + λb₁ and r = a₂ + μb₂ isRead the answer and worked solution →Two lines r = a₁ + λb₁ and r = a₂ + μb₂ are coplanar if and only ifRead the answer and worked solution →The equation of a plane passing through point (x₁, y₁, z₁) with normal vector (a, b, c) isRead the answer and worked solution →The angle between the lines with direction ratios (1, 2, 2) and (2, 1, 2) satisfiesRead the answer and worked solution →The equation of the plane through the points (1,0,0), (0,1,0), (0,0,1) isRead the answer and worked solution →The distance between the parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z = 5 isRead the answer and worked solution →The foot of the perpendicular from origin to the plane 2x + 3y - 6z = 14 isRead the answer and worked solution →True or false: direction cosinesRead the answer and worked solution →Order the steps: direction ratios to direction cosinesRead the answer and worked solution →Complete the definition of direction cosinesRead the answer and worked solution →True or false: angles between linesRead the answer and worked solution →Assertion and reasonRead the answer and worked solution →Two are true, one is notRead the answer and worked solution →True or false: reading the chapter's scopeRead the answer and worked solution →Match each condition to what it meansRead the answer and worked solution →Assertion and reason: equations of a lineRead the answer and worked solution →Two lines with direction vectors b₁ and b₂ are parallel if and only ifRead the answer and worked solution →Direction cosines or direction ratios?Read the answer and worked solution →Flip and pair: symbol with meaningRead the answer and worked solution →The equation of a plane in normal form (with unit normal n̂ and perpendicular distance d from origin) isRead the answer and worked solution →Match each equation to what it describesRead the answer and worked solution →Match each term to its definitionRead the answer and worked solution →The equation of a plane parallel to x + 2y - 3z = 5 and passing through (1, 1, 1) isRead the answer and worked solution →

Worked practice questions

The vector equation of a line passing through point a and parallel to vector b is

  1. r = a × b
  2. r = a · b
  3. r = a + λb
  4. r = λa + b
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Correct answer: C. r = a + λb

1. A line through point a in the direction of b is the set of all r reachable by moving from a along b. 2. The position vector is r = a + λb, where λ is the scalar parameter. 3. Varying λ traces the entire line. 4. So option C is correct. _Source: syllabus Maths Part 2 Ch 11 "Three Dimensional Geometry", §11.3.1_

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The equation of a plane through the intersection of π₁ = 0 and π₂ = 0 has the form

  1. π₁ · π₂ = 0
  2. π₁ + λπ₂ = 0
  3. π₁ × π₂ = 0
  4. (π₁ - π₂) · λ = 0
Show answer and solution

Correct answer: B. π₁ + λπ₂ = 0

1. Let π₁ and π₂ be linear expressions in x, y, z each equal to zero on its plane. 2. Their linear combination π₁ + λπ₂ vanishes at any common point of the two planes. 3. Varying λ traces the pencil of planes through the intersection line. 4. So the family is π₁ + λπ₂ = 0. _Source: syllabus Maths Part 2 Ch 11 "Three Dimensional Geometry", §11.6.5_

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The line r = (2, -1, 4) + λ(3, 0, 2), expressed in symmetric Cartesian form, is

  1. (x-2)/3 = (y+1)/0 = (z-4)/2
  2. (x+2)/3 = (y-1)/0 = (z+4)/2
  3. (x-2)/3 = y = (z-4)/2
  4. (x-3)/2 = y = (z-2)/4
Show answer and solution

Correct answer: A. (x-2)/3 = (y+1)/0 = (z-4)/2

1. The line passes through (2, -1, 4) with direction (3, 0, 2). 2. Symmetric form: (x - x₀)/a = (y - y₀)/b = (z - z₀)/c. 3. Substitute: (x-2)/3 = (y+1)/0 = (z-4)/2. 4. The middle term with denominator 0 encodes y = -1 in formal notation. _Source: syllabus Maths Part 2 Ch 11 "Three Dimensional Geometry", §11.3.1_

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