Unit 3: Geometry and Trigonometry
Topic 3.17: Vector planes (HL only) Questions
Practice 20 exam-style questions for IB Math AA SL Topic 3.17. Review the question stems below, then unlock the full Question Bank to access markschemes, model answers, and AI grading.
11 mark
2026
The plane r·(1, 2, −2) = 6 in Cartesian form is:
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2026
The normal vector of the plane 3x − 4y + z = 7 is:
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A plane with normal (1, 1, 1) passes through (2, 0, 1). Its constant d is:
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A plane passes through A(2, −1, 3) and has normal n = (1, 2, −1). Find its equation in the form r·n = a·n.
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Three points A, B, C define a plane provided:
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Show that the point P(2, 1, 4) lies on the plane 2x − 3y + z = 5, but Q(1, 1, 1) does not.
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Convert the plane r·(2, −1, 3) = 7 into Cartesian form, and state whether (1, 1, 2) lies on it.
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2026
A plane has Cartesian equation 5x − y + 4z = 12. State (i) a normal vector and (ii) the value of d. Then verify (1, −3, 1) lies on it.
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2026
To write a plane in Cartesian form once you have the normal n = (a, b, c), you still need:
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A plane passes through the origin and has normal n = (2, −5, 1). Find its Cartesian equation.
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Find the Cartesian equation of the plane.
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The planes x + 2y − z = 3 and 2x + 4y − 2z = 9 are:
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If AB × AC = (0, 0, 0) for three points A, B, C, then:
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A plane contains the line r = (0, 1, 2) + λ(1, 0, 2) and the point P(2, 3, 1). Find a normal to the plane.
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A plane has normal n = (3, 0, −2) and contains B(1, 5, 2). Find its Cartesian equation.
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Find the value of k so that the point (k, 2, −1) lies on the plane 2x + y − 3z = 9.
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Find the equation of the plane through D(2, 0, 1), E(2, 1, 3) and F(4, 0, 1) in the form r·n = k.
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Given AB = (1, 0, 1) and AC = (0, 1, 1), the normal AB × AC is:
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Show that is normal to the plane.
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