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AP Intermediate 2nd Year Vector Algebra and 3D Geometry — practice questions — page 7

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 181–210 of 218.

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 →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 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 →

Worked practice questions

True or false: direction cosines

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Questions
  1. Statement is true
    Yes
    Statement
    For any line, l² + m² + n² = 1.
  2. Statement is true
    No
    Statement
    Direction ratios must also satisfy a² + b² + c² = 1.
  3. Statement is true
    Yes
    Statement
    Direction cosines are the cosines of the angles made with the axes.
  4. Statement is true
    No
    Statement
    A line has exactly one set of direction ratios.
  5. Statement is true
    Yes
    Statement
    Direction ratios are proportional to the direction cosines.
Direction cosines are normalised, direction ratios are not. That single difference decides most questions here. Source: syllabus Maths Part II, Ch 11 Three Dimensional Geometry.

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Order the steps: direction ratios to direction cosines

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Steps
  1. Start from the direction ratios a, b, c
  2. Square each of them and add the three squares
  3. Take the square root of that sum
  4. Divide each of a, b and c by that root
  5. The results are the direction cosines l, m, n
Your syllabus gives l = a/√(a²+b²+c²) and similarly for m and n, which is exactly this normalising procedure. Source: syllabus Maths Part II, Ch 11 Three Dimensional Geometry.

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Complete the definition of direction cosines

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Missing parts
  1. Answer
    cosines
    Choices
    1. cosines
    2. sines
    3. tangents
  2. Answer
    coordinate axes
    Choices
    1. coordinate axes
    2. coordinate planes
    3. position vectors
  3. Answer
    l² + m² + n² = 1
    Choices
    1. l² + m² + n² = 1
    2. l + m + n = 1
    3. lmn = 1
Complete the statement
Direction cosines of a line are the {0} of the angles the line makes with the positive directions of the {1}. They always satisfy {2}.
Word for word from the opening bullet of the summary. Source: syllabus Maths Part II, Ch 11 Three Dimensional Geometry.

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