IB Mathematics: Applications and Interpretation Quiz: Vector Lines And Planes
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Vector Lines And PlanesQuestion 1 of 8

A flat hillside is modelled by the plane with equation 5x + 2y + z = 100. A local coordinate system is used where the positive x-axis points East, the positive y-axis points North, and the z-axis points vertically upwards.

In this model, what is the compass direction of steepest ascent on the hillside?

A direction mostly South and slightly West.
A direction mostly West and slightly South.
A direction mostly North and slightly East.
A direction mostly East and slightly North.
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IB Mathematics: Applications and Interpretation Quiz

IB Mathematics: Applications and Interpretation Quiz: Vector Lines And Planes

Practice Vector Lines And Planes in IB Mathematics: Applications and Interpretation with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Vector Lines And Planes, giving you a quick way to practice the rules, question types, and explanations that matter most for IB Mathematics: Applications and Interpretation.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A flat hillside is modelled by the plane with equation 5x + 2y + z = 100. A local coordinate system is used where the positive x-axis points East, the positive y-axis points North, and the z-axis points vertically upwards.

In this model, what is the compass direction of steepest ascent on the hillside?

  1. A direction mostly South and slightly West.
  2. A direction mostly West and slightly South.
  3. A direction mostly North and slightly East.
  4. A direction mostly East and slightly North. (correct answer)
Explanation: The gradient of a plane f(x,y,z)=c is given by the normal vector. The direction of steepest ascent on the surface z = f(x,y) = 100 - 5x - 2y is given by the vector (-∂z/∂x, -∂z/∂y), which corresponds to the projection of the normal vector onto the xy-plane. The normal vector to the plane 5x + 2y + z = 100 is n = (5, 2, 1). The projection of this vector onto the xy-plane is (5, 2, 0) or simply (5, 2). This vector represents the direction of the greatest rate of change of height, i.e., the direction of steepest ascent. A vector (5, 2) in the xy-plane corresponds to moving 5 units in the positive x-direction (East) and 2 units in the positive y-direction (North). This is a direction that is mostly East and slightly North.

Question 2

A scientist models the positions of three satellites A, B, and C with coordinates A(1, 2, 3), B(3, 5, 7), and C(7, 11, 15). For a signal relay to work, the satellites must be collinear.

Which statement correctly describes the alignment of the satellites?

  1. Satellite B is equidistant from satellites A and C.
  2. The satellites form the vertices of a right-angled triangle.
  3. The vectors AB and BC are perpendicular.
  4. The satellites lie on the same straight line path. (correct answer)
Explanation: To check for collinearity, we find the vectors between the points and see if they are parallel. Vector AB = B - A = (3-1, 5-2, 7-3) = (2, 3, 4). Vector BC = C - B = (7-3, 11-5, 15-7) = (4, 6, 8). We can see that BC = 2 * AB. Since one vector is a scalar multiple of the other and they share a common point (B), the points A, B, and C are collinear. Therefore, the satellites lie on the same straight line path. A is incorrect because |BC| = 2|AB|. The magnitude of AB is √29 and the magnitude of BC is √116 = 2√29. C is incorrect as the vectors are parallel, not perpendicular.

Question 3

A scientist models the positions of three satellites A, B, and C with coordinates A(1, 2, 3), B(3, 5, 7), and C(7, 11, 15). For a signal relay to work, the satellites must be collinear.

Which statement correctly describes the alignment of the satellites?

  1. Satellite B is equidistant from satellites A and C.
  2. The satellites form the vertices of a right-angled triangle.
  3. The vectors AB and BC are perpendicular.
  4. The satellites lie on the same straight line path. (correct answer)
Explanation: To check for collinearity, we find the vectors between the points and see if they are parallel. Vector AB = B - A = (3-1, 5-2, 7-3) = (2, 3, 4). Vector BC = C - B = (7-3, 11-5, 15-7) = (4, 6, 8). We can see that BC = 2 * AB. Since one vector is a scalar multiple of the other and they share a common point (B), the points A, B, and C are collinear. Therefore, the satellites lie on the same straight line path. A is incorrect because |BC| = 2|AB|. The magnitude of AB is √29 and the magnitude of BC is √116 = 2√29. C is incorrect as the vectors are parallel, not perpendicular.

Question 4

A force, represented by the vector F = (5, 2, -1) Newtons, is applied to an object. The object is constrained to move along a track with direction vector d = (1, 1, 1).

What is the magnitude of the component of the force that acts along the track?

  1. 1.73 N
  2. 3.46 N (correct answer)
  3. 5.48 N
  4. 6.00 N
Explanation: The component of force F along direction d has magnitude given by the scalar projection formula: |(F ⋅ d)|/|d| Calculate the dot product: F ⋅ d = (5)(1) + (2)(1) + (-1)(1) = 5 + 2 - 1 = 6 Calculate |d| = √(1² + 1² + 1²) = √3 ≈ 1.732 Component magnitude = 6/√3 = 6/1.732 ≈ 3.46 N Option A gives |d| only. Option C gives |F| = √(25+4+1) = √30 ≈ 5.48. Option D gives the dot product value before division.

Question 5

A flat hillside is modelled by the plane with equation 5x + 2y + z = 100. A local coordinate system is used where the positive x-axis points East, the positive y-axis points North, and the z-axis points vertically upwards.

In this model, what is the compass direction of steepest ascent on the hillside?

  1. A direction mostly South and slightly West.
  2. A direction mostly West and slightly South.
  3. A direction mostly North and slightly East.
  4. A direction mostly East and slightly North. (correct answer)
Explanation: The gradient of a plane f(x,y,z)=c is given by the normal vector. The direction of steepest ascent on the surface z = f(x,y) = 100 - 5x - 2y is given by the vector (-∂z/∂x, -∂z/∂y), which corresponds to the projection of the normal vector onto the xy-plane. The normal vector to the plane 5x + 2y + z = 100 is n = (5, 2, 1). The projection of this vector onto the xy-plane is (5, 2, 0) or simply (5, 2). This vector represents the direction of the greatest rate of change of height, i.e., the direction of steepest ascent. A vector (5, 2) in the xy-plane corresponds to moving 5 units in the positive x-direction (East) and 2 units in the positive y-direction (North). This is a direction that is mostly East and slightly North.

Question 6

The corner of a room is formed by the intersection of three planes representing the walls and floor: P₁: x + z = 5 P₂: y + z = 6 P₃: x + y = 7

Find the coordinates of the corner point where the three planes intersect.

  1. (2, 5, 3)
  2. (3, 4, 2) (correct answer)
  3. (4, 3, 1)
  4. (1, 6, 4)
Explanation: We solve the system of three linear equations:
  1. x + z = 5
  2. y + z = 6
  3. x + y = 7
From equation (1): z = 5 - x From equation (2): z = 6 - y Setting these equal: 5 - x = 6 - y, which gives y = x + 1 Substituting into equation (3): x + (x + 1) = 7 2x + 1 = 7, so x = 3 Then y = 3 + 1 = 4 And z = 5 - 3 = 2 The intersection point is (3, 4, 2). Option A results from solving incorrectly. Option C swaps x and y values. Option D represents a common algebraic error in the substitution step.

Question 7

A force, represented by the vector F = (5, 2, -1) Newtons, is applied to an object. The object is constrained to move along a track with direction vector d = (1, 1, 1).

What is the magnitude of the component of the force that acts along the track?

  1. 1.73 N
  2. 3.46 N (correct answer)
  3. 5.48 N
  4. 6.00 N
Explanation: The component of force F along direction d has magnitude given by the scalar projection formula: |(F ⋅ d)|/|d| Calculate the dot product: F ⋅ d = (5)(1) + (2)(1) + (-1)(1) = 5 + 2 - 1 = 6 Calculate |d| = √(1² + 1² + 1²) = √3 ≈ 1.732 Component magnitude = 6/√3 = 6/1.732 ≈ 3.46 N Option A gives |d| only. Option C gives |F| = √(25+4+1) = √30 ≈ 5.48. Option D gives the dot product value before division.

Question 8

The corner of a room is formed by the intersection of three planes representing the walls and floor: P₁: x + z = 5 P₂: y + z = 6 P₃: x + y = 7

Find the coordinates of the corner point where the three planes intersect.

  1. (2, 5, 3)
  2. (3, 4, 2) (correct answer)
  3. (4, 3, 1)
  4. (1, 6, 4)
Explanation: We solve the system of three linear equations:
  1. x + z = 5
  2. y + z = 6
  3. x + y = 7
From equation (1): z = 5 - x From equation (2): z = 6 - y Setting these equal: 5 - x = 6 - y, which gives y = x + 1 Substituting into equation (3): x + (x + 1) = 7 2x + 1 = 7, so x = 3 Then y = 3 + 1 = 4 And z = 5 - 3 = 2 The intersection point is (3, 4, 2). Option A results from solving incorrectly. Option C swaps x and y values. Option D represents a common algebraic error in the substitution step.