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Indian Navy MR Three Dimensional Geometry — practice questions

Explore 26 free Three Dimensional Geometry questions in Indian Navy MR Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Three Dimensional Geometry questions and exercises with answers

Showing questions 1–26 of 26.

A line passes through the points $(1, 2, 3)$ and $(4, 6, 9)$. The direction ratios of the line are:Read the answer and worked solution →Match each term to its definitionRead the answer and worked solution →True or false: direction cosinesRead the answer and worked solution →Direction cosines or direction ratios?Read the answer and worked solution →Order the steps: direction ratios to direction cosinesRead 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 →Match each equation to what it describesRead the answer and worked solution →Complete the definition of direction cosinesRead the answer and worked solution →Complete the definition of skew linesRead the answer and worked solution →True or false: angles between linesRead the answer and worked solution →Assertion and reasonRead the answer and worked solution →Spot the one that does not belongRead the answer and worked solution →Two are true, one is notRead the answer and worked solution →Reading the equationsRead the answer and worked solution →Flip and pair: symbol with meaningRead the answer and worked solution →Same plane or different planes?Read the answer and worked solution →Three of these are wrong. Find the one that holds.Read the answer and worked solution →Order the steps: angle between two lines from direction ratiosRead the answer and worked solution →Tap each statement that is true of a line's equationRead 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 →The normalising relationRead the answer and worked solution →Spot the odd one outRead the answer and worked solution →Assertion and reason: equations of a lineRead the answer and worked solution →Which quantity is which?Read the answer and worked solution →

Worked practice questions

A line passes through the points $(1, 2, 3)$ and $(4, 6, 9)$. The direction ratios of the line are:

  1. $(4, 6, 9)$
  2. $(3, 4, 6)$
  3. $(5, 8, 12)$
  4. $(1, 2, 3)$
Show answer and solution

Correct answer: B. $(3, 4, 6)$

Direction ratios of the line joining two points are the differences in coordinates: $(x_2 - x_1, y_2 - y_1, z_2 - z_1)$. Here: $(4 - 1, 6 - 2, 9 - 3) = (3, 4, 6)$. Direction ratios are not unique. $(6, 8, 12)$ or $(1.5, 2, 3)$ would also be valid since multiplying by any non-zero scalar preserves the direction. The standard answer just uses the raw differences. Direction cosines (the normalised version) are $(3, 4, 6) / \sqrt{9 + 16 + 36} = (3, 4, 6) / \sqrt{61}$.

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Match each term to its definition

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Matching pairs
  1. Left
    Direction cosines
    Right
    cosines of the angles a line makes with the axes
  2. Left
    Direction ratios
    Right
    numbers proportional to the direction cosines
  3. Left
    Skew lines
    Right
    lines that neither meet nor run parallel
  4. Left
    l² + m² + n²
    Right
    always equals 1
The chapter's four core definitions, taken straight from the summary. Source: syllabus Maths Part II, Ch 11 Three Dimensional Geometry.

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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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