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AP Intermediate 2nd Year Vector Algebra and 3D Geometry — practice questions

Explore 218 free Vector Algebra and 3D Geometry questions in AP Intermediate 2nd Year 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 1–30 of 218.

Three vectors $\vec a, \vec b, \vec c$ are coplanar if and only if:Read the answer and worked solution →If l, m, n are the direction cosines of a line in space, thenRead the answer and worked solution →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 in normal form (with unit normal n̂ and perpendicular distance d from origin) isRead the answer and worked solution →A plane meets the axes at (a,0,0), (0,b,0), (0,0,c). Its equation isRead 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 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 angle between two planes with normals n₁ and n₂ is given byRead the answer and worked solution →The angle ϕ between a line with direction b and a plane with normal n satisfiesRead 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 →Two lines with direction vectors b₁ and b₂ are parallel if and only ifRead 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 perpendicular distance from a point P with position vector p to a line r = a + λb 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 distance of the point (2, 1, -1) from the plane 2x - y + 2z + 5 = 0 isRead 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 →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 line r = (2, -1, 4) + λ(3, 0, 2), expressed in symmetric Cartesian form, isRead the answer and worked solution →The shortest distance between r = (1,2,3) + λ(2,3,4) and r = (2,4,5) + μ(3,4,5) isRead 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 →The plane through the points (1,1,0), (1,0,1) and (0,1,1) has equationRead the answer and worked solution →A line has direction ratios (1, 1, 2) and meets the plane x - y + z = 0. The angle ϕ between them satisfiesRead 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 foot of the perpendicular from origin to the plane 2x + 3y - 6z = 14 isRead 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 →

Worked practice questions

Three vectors $\vec a, \vec b, \vec c$ are coplanar if and only if:

  1. $\vec a + \vec b + \vec c = 0$ on the school chart here
  2. $\vec a\cdot\vec b = \vec b\cdot\vec c = 0$ on the chart all the time
  3. $|\vec a| + |\vec b| = |\vec c|$ on chart at all times
  4. Their scalar triple product $[\vec a\, \vec b\, \vec c] = 0$
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Correct answer: D. Their scalar triple product $[\vec a\, \vec b\, \vec c] = 0$

Three vectors are coplanar iff their scalar triple product (parallelepiped volume) is zero.

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If l, m, n are the direction cosines of a line in space, then

  1. l + m + n = 1
  2. l² + m² + n² = 1
  3. lmn = 1
  4. l² + m² + n² = 0
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Correct answer: B. l² + m² + n² = 1

1. Direction cosines are the cosines of angles a line makes with the x, y, z axes. 2. If the line has unit direction vector (l, m, n), its squared length equals one. 3. So l² + m² + n² = 1. 4. This is the standard identity for direction cosines. _Source: syllabus Maths Part 2 Ch 11 "Three Dimensional Geometry", §11.2.1_

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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
Show answer and solution

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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