Mathematics: Applications & Interpretation HL
Geometry and trigonometry: vectors — Topic 3 HL
- 1.
A line has vector equation r = (1, 2, 3) + t(2, −1, 1). Find the coordinates of the point on the line when t = 2.
[2 marks] - 2.
Marking analysis: A learner attempts the following task: “A line has vector equation r = (1, 2, 3) + t(2, −1, 1). Find the coordinates of the point on the line when t = 2.” Their response addresses only this point: “Substitutes t = 2 into each component: x = 1 + 2(2), y = 2 − 2, z = 3 + 2.” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] - 3.
Find the angle between the vectors a = (3, 4, 0) and b = (1, 2, 2), using the dot product.
[4 marks] - 4.
Marking analysis: A learner attempts the following task: “Find the angle between the vectors a = (3, 4, 0) and b = (1, 2, 2), using the dot product.” Their response addresses only this point: “Calculates a·b = (3)(1) + (4)(2) + (0)(2) = 11.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 5.
Find the cross product a × b for a = (2, 0, 0) and b = (1, 3, 0), and hence find the area of the parallelogram formed by these two vectors.
[4 marks] - 6.
Marking analysis: A learner attempts the following task: “Find the cross product a × b for a = (2, 0, 0) and b = (1, 3, 0), and hence find the area of the parallelogram formed by these two vectors.” Their response addresses only this point: “Uses the cross product formula a × b = (a2b3−a3b2, a3b1−a1b3, a1b2−a2b1).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 7.
Triangle ABC has vertices A(0, 0, 0), B(4, 0, 0), C(0, 3, 0). Use the cross product of vectors AB and AC to find the area of the triangle.
[4 marks] - 8.
Marking analysis: A learner attempts the following task: “Triangle ABC has vertices A(0, 0, 0), B(4, 0, 0), C(0, 3, 0). Use the cross product of vectors AB and AC to find the area of the triangle.” Their response addresses only this point: “Finds AB = (4, 0, 0) and AC = (0, 3, 0).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 9.
Find the scalar projection of vector a = (3, 4) onto vector b = (1, 0).
[3 marks] - 10.
Marking analysis: A learner attempts the following task: “Find the scalar projection of vector a = (3, 4) onto vector b = (1, 0).” Their response addresses only this point: “Uses the scalar projection formula: (a·b)/|b|.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 11.
Find the distance between the points A(1, 2, 3) and B(4, 6, 3) in three-dimensional space.
[2 marks] - 12.
Marking analysis: A learner attempts the following task: “Find the distance between the points A(1, 2, 3) and B(4, 6, 3) in three-dimensional space.” Their response addresses only this point: “Uses the 3D distance formula √[(x2−x1)² + (y2−y1)² + (z2−z1)²].” Evaluate the response against the complete 2-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[2 marks] - 13.
Line 1 has vector equation r = (1, 1, 0) + t(1, 2, 0). Line 2 has vector equation r = (0, 3, 0) + s(1, −1, 0). Determine whether the two lines intersect, and if so, find the point of intersection.
[5 marks] - 14.
Marking analysis: A learner attempts the following task: “Line 1 has vector equation r = (1, 1, 0) + t(1, 2, 0). Line 2 has vector equation r = (0, 3, 0) + s(1, −1, 0). Determine whether the two lines intersect, and if so, find the point of intersection.” Their response addresses only this point: “Sets the x-components equal: 1 + t = s.” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks] - 15.
Two sites A(2, 3) and B(8, 7) are used to generate part of a Voronoi diagram. Find the equation of the perpendicular bisector of AB, which forms the boundary between the two sites' regions.
[5 marks] - 16.
Marking analysis: A learner attempts the following task: “Two sites A(2, 3) and B(8, 7) are used to generate part of a Voronoi diagram. Find the equation of the perpendicular bisector of AB, which forms the boundary between the two sites' regions.” Their response addresses only this point: “Finds the midpoint of AB: ((2+8)/2, (3+7)/2) = (5, 5).” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks] - 17.
A support cable has direction vector (1, 2, 2) and is attached to a horizontal floor, whose normal vector is (0, 0, 1). Find the angle the cable makes with the floor.
[4 marks] - 18.
Marking analysis: A learner attempts the following task: “A support cable has direction vector (1, 2, 2) and is attached to a horizontal floor, whose normal vector is (0, 0, 1). Find the angle the cable makes with the floor.” Their response addresses only this point: “Finds the angle between the cable's direction and the normal vector using the dot product: cos(φ) = |d·n|/(|d||n|).” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 19.
Find the shortest distance from the point P(1, 1, 1) to the line through A(0, 0, 0) with direction vector d = (1, 1, 0).
[5 marks] - 20.
Marking analysis: A learner attempts the following task: “Find the shortest distance from the point P(1, 1, 1) to the line through A(0, 0, 0) with direction vector d = (1, 1, 0).” Their response addresses only this point: “Finds the vector AP = (1, 1, 1).” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks] - 21.
A cuboid has dimensions 6 cm × 8 cm × 10 cm. Find the angle between the space diagonal of the cuboid and its base.
[4 marks] - 22.
Marking analysis: A learner attempts the following task: “A cuboid has dimensions 6 cm × 8 cm × 10 cm. Find the angle between the space diagonal of the cuboid and its base.” Their response addresses only this point: “Finds the base diagonal: √(6² + 8²) = 10 cm.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 23.
Three mobile phone towers are located at A(0, 0), B(6, 0), and C(3, 5). A phone at point P(2, 2) connects to whichever tower is closest. Calculate the distance from P to each tower, and state which tower the phone connects to.
[4 marks] - 24.
Marking analysis: A learner attempts the following task: “Three mobile phone towers are located at A(0, 0), B(6, 0), and C(3, 5). A phone at point P(2, 2) connects to whichever tower is closest. Calculate the distance from P to each tower, and state which tower the phone connects to.” Their response addresses only this point: “Calculates PA = √(2² + 2²) ≈ 2.83.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 25.
A line has vector equation r = (1, 2, 3) + t(2, −1, 1). Determine whether the point (7, −1, 7) lies on this line.
[4 marks] - 26.
Marking analysis: A learner attempts the following task: “A line has vector equation r = (1, 2, 3) + t(2, −1, 1). Determine whether the point (7, −1, 7) lies on this line.” Their response addresses only this point: “Sets up three equations: 1 + 2t = 7, 2 − t = −1, 3 + t = 7.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 27.
Find the unit vector in the direction of v = (3, −4, 12).
[3 marks] - 28.
Marking analysis: A learner attempts the following task: “Find the unit vector in the direction of v = (3, −4, 12).” Their response addresses only this point: “Calculates the magnitude |v| = √(3² + (−4)² + 12²) = √169 = 13.” Evaluate the response against the complete 3-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[3 marks] - 29.
Find the angle between two lines with direction vectors d1 = (1, 1, 1) and d2 = (2, −1, 1).
[4 marks] - 30.
Marking analysis: A learner attempts the following task: “Find the angle between two lines with direction vectors d1 = (1, 1, 1) and d2 = (2, −1, 1).” Their response addresses only this point: “Calculates d1·d2 = (1)(2) + (1)(−1) + (1)(1) = 2.” Evaluate the response against the complete 4-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[4 marks] - 31.
A boat has a velocity of (6, 0) km/h relative to the water (heading due east), while a current has velocity (1, 2) km/h. Find the boat's resultant velocity vector relative to the ground, its speed, and its bearing.
[5 marks] - 32.
Marking analysis: A learner attempts the following task: “A boat has a velocity of (6, 0) km/h relative to the water (heading due east), while a current has velocity (1, 2) km/h. Find the boat's resultant velocity vector relative to the ground, its speed, and its bearing.” Their response addresses only this point: “Adds the velocity vectors: (6, 0) + (1, 2) = (7, 2).” Evaluate the response against the complete 5-mark task. Identify what earns credit and state every additional requirement needed for full marks.
[5 marks]