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JEE Main Vector Algebra and 3D Geometry — practice questions — page 4

Explore 126 free Vector Algebra and 3D Geometry questions in JEE Main 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 91–120 of 126.

Two non-zero vectors a and b are PERPENDICULAR if and only if:Read the answer and worked solution →If a = 2i + 3j + k and b = i − j + 2k, then a · b equals:Read the answer and worked solution →Reading the equationsRead the answer and worked solution →Same plane or different planes?Read the answer and worked solution →Spot the one that does not belongRead the answer and worked solution →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 →Direction cosines or direction ratios?Read the answer and worked solution →Flip and pair: symbol with meaningRead 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 angle θ between two PLANES with normals n1 and n2 satisfies:Read the answer and worked solution →The vector triple product a × (b × c) simplifies via:Read the answer and worked solution →The Cartesian equation of a plane with intercepts 2, 3, 6 on the x, y, z axes is:Read the answer and worked solution →For direction cosines l, m, n of any line in 3D:Read the answer and worked solution →The normalising relationRead the answer and worked solution →The scalar triple product [a b c] geometrically represents:Read the answer and worked solution →Scalar or vector?Read the answer and worked solution →Three of these are wrong. Find the one that holds.Read the answer and worked solution →Flip and pair: symbol with meaningRead 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 →Tap each statement that is true of a line's equationRead the answer and worked solution →

Worked practice questions

Two non-zero vectors a and b are PERPENDICULAR if and only if:

  1. $a + b = 0$ (zero sum)
  2. $a × b = 0$ (zero cross)
  3. $a · b = 0$ (zero dot product)
  4. $|a| = |b|$ (equal magnitude)
Show answer and solution

Correct answer: C. $a · b = 0$ (zero dot product)

PERPENDICULAR vectors have dot product = |a||b| cos 90° = 0. So a · b = 0. CROSS PRODUCT = 0 means PARALLEL vectors.

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If a = 2i + 3j + k and b = i − j + 2k, then a · b equals:

  1. $6$
  2. $1$
  3. $3$
  4. $5$
Show answer and solution

Correct answer: B. $1$

We have a · b = (2)(1) + (3)(-1) + (1)(2). That is a · b = 2 − 3 + 2. So a · b = 1.

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Reading the equations

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Questions
  1. Question
    In r = a + λb, the vector b gives
    Choices
    1. the direction of the line
    2. a point on the line
    3. the length
    Correct choice
    the direction of the line
  2. Question
    In r = a + λb, the vector a gives
    Choices
    1. the direction
    2. the position vector of a point on the line
    3. the angle
    Correct choice
    the position vector of a point on the line
  3. Question
    Which form passes through two given points?
    Choices
    1. r = a + λb
    2. l² + m² + n² = 1
    3. r = a + λ(b − a)
    Correct choice
    r = a + λ(b − a)
  4. Question
    In the Cartesian form, the denominators l, m, n are
    Choices
    1. direction cosines
    2. coordinates
    3. angles
    Correct choice
    direction cosines
Recognising which form answers which question is most of the work in this chapter. Source: syllabus Maths Part II, Ch 11 Three Dimensional Geometry.

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