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GRE Vector Algebra and 3D Geometry — practice questions

Explore 18 free Vector Algebra and 3D Geometry questions in GRE 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 1–18 of 18.

Three vectors $\vec a, \vec b, \vec c$ are coplanar 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 →Two non-zero vectors a and b are PERPENDICULAR if and only if:Read the answer and worked solution →A UNIT VECTOR in the direction of v = 3i + 4j is:Read the answer and worked solution →The scalar triple product [a b c] geometrically represents: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 angle θ between two PLANES with normals n1 and n2 satisfies:Read the answer and worked solution →Three vectors $\vec a, \vec b, \vec c$ are coplanar if and only if:Read the answer and worked solution →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 →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 scalar triple product [a b c] geometrically represents:Read the answer and worked solution →A UNIT VECTOR in the direction of v = 3i + 4j is:Read the answer and worked solution →

Worked practice questions

Three vectors $\vec a, \vec b, \vec c$ are coplanar if and only if:

  1. $\vec a + \vec b + \vec c = 0$ on the school chart here
  2. $\vec a\cdot\vec b = \vec b\cdot\vec c = 0$ on the chart all the time
  3. $|\vec a| + |\vec b| = |\vec c|$ on chart at all times
  4. Their scalar triple product $[\vec a\, \vec b\, \vec c] = 0$
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Correct answer: D. Their scalar triple product $[\vec a\, \vec b\, \vec c] = 0$

Three vectors are coplanar iff their scalar triple product (parallelepiped volume) is zero.

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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. $5$
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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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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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