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Indian Navy 10+2 B.Tech Cadet Three Dimensional Geometry — practice questions

Explore 28 free Three Dimensional Geometry questions in Indian Navy 10+2 B.Tech Cadet 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 1–28 of 28.

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 →Two lines are parallel if their direction ratios are:Read the answer and worked solution →Equation of straight line passing through two points A(1,2,3) and B(4,5,6):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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Two lines are parallel if their direction ratios are:

  1. Perpendicular
  2. Proportional (a₁/a₂ = b₁/b₂ = c₁/c₂)
  3. Random
  4. Equal
Show answer and solution

Correct answer: B. Proportional (a₁/a₂ = b₁/b₂ = c₁/c₂)

Two lines parallel iff their direction ratios are proportional. Two lines perpendicular iff dot product of direction vectors = 0: a₁a₂ + b₁b₂ + c₁c₂ = 0.

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Equation of straight line passing through two points A(1,2,3) and B(4,5,6):

  1. (x-1)/3 = (y-2)/3 = (z-3)/3 (direction = B-A = (3,3,3))
  2. x = y = z
  3. (x-1)/4 = (y-2)/5 = (z-3)/6
  4. x + y + z = 6
Show answer and solution

Correct answer: A. (x-1)/3 = (y-2)/3 = (z-3)/3 (direction = B-A = (3,3,3))

Direction = B - A = (3, 3, 3) or any scalar multiple (1, 1, 1). Through A(1,2,3): (x-1)/1 = (y-2)/1 = (z-3)/1, equivalently (x-1)/3 = (y-2)/3 = (z-3)/3.

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More Mathematics topics

SequencesBinomial TheoremGeometryProbability & StatisticsSequences and SeriesVectors & 3D GeometryPermutations and CombinationsLinear Programming