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AP Intermediate 2nd Year Vector Algebra and 3D Geometry — practice questions — page 8

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 211–218 of 218.

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 →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

Tap each statement that is true of a line's equation

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Questions
  1. Items
    1. Text
      b sets the direction
      Matches the category
      Yes
    2. Text
      a is a point on the line
      Matches the category
      Yes
    3. Text
      λ sweeps along the line
      Matches the category
      Yes
    4. Text
      It describes a plane
      Matches the category
      No
    5. Text
      b must be a unit vector
      Matches the category
      No
    Category
    True of r = a + λb
  2. Items
    1. Text
      (x₁, y₁, z₁) lies on the line
      Matches the category
      Yes
    2. Text
      l, m, n are direction cosines
      Matches the category
      Yes
    3. Text
      It needs two points
      Matches the category
      No
    4. Text
      It only works in two dimensions
      Matches the category
      No
    Category
    True of the Cartesian form (x−x₁)/l = (y−y₁)/m = (z−z₁)/n
Sorting what the vector form does and does not tell you. Source: syllabus Maths Part II, Ch 11 Three Dimensional Geometry.

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The angle between the planes 2x - y + 2z = 5 and x + 2y - 2z = 4 satisfies

  1. cos θ = -4/9
  2. cos θ = 4/9
  3. cos θ = -2/3
  4. cos θ = 0
Show answer and solution

Correct answer: A. cos θ = -4/9

1. Normal to first plane: (2, -1, 2); magnitude √(4+1+4) = 3. 2. Normal to second plane: (1, 2, -2); magnitude √(1+4+4) = 3. 3. Dot product of normals: 2(1) + (-1)(2) + 2(-2) = -4. 4. cos θ = -4 / (3·3) = -4/9. _Source: syllabus Maths Part 2 Ch 11 "Three Dimensional Geometry", §11.8_

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Skew lines

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Questions
  1. Question
    Skew lines are
    Choices
    1. parallel but not equal
    2. intersecting at right angles
    3. neither parallel nor intersecting
    Correct choice
    neither parallel nor intersecting
  2. Question
    Two skew lines lie
    Choices
    1. in different planes
    2. in the same plane
    3. always on the x-axis
    Correct choice
    in different planes
  3. Question
    Can two skew lines ever meet?
    Choices
    1. Yes, at one point
    2. No
    3. Only at the origin
    Correct choice
    No
  4. Question
    The angle between skew lines is measured using
    Choices
    1. the distance between them
    2. their midpoints
    3. intersecting lines drawn parallel to each
    Correct choice
    intersecting lines drawn parallel to each
Your syllabus defines skew lines as lines in space that are neither parallel nor intersecting, and notes they lie in different planes. Source: syllabus Maths Part II, Ch 11 Three Dimensional Geometry.

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