Position Vectors and Displacement
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Position Vectors and Displacement — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionFinding Displacement Between Two PointsAssessment Practice
5 marks~8 minCriterion B
The diagram shows the first four figures of a pattern made from dots arranged in an L-shape.

Figure 1 has 2 dots, Figure 2 has 5 dots, Figure 3 has 10 dots, and Figure 4 has 17 dots.
a
Write down the number of dots in Figure 5. [1]
b
Find a rule for the number of dots DD in Figure nn, in the form D=n2+cD = n^2 + c, where cc is a constant to be determined. [2]
c
Verify your rule for Figure 3. Justify why the rule must contain an n2n^2 term by referring to the structure of the L-shape. [2]
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Solutions

2QuestionDefinition and Use of Position VectorsAssessment Practice
7 marks~11 minCriterion A
Point AA has coordinates (3,−2)(3, -2) and point BB has coordinates (−1,4)(-1, 4).
a
Calculate the exact length of ABAB, giving your answer as a simplified surd. [2]
b
Find the coordinates of the point MM that divides ABAB in the ratio 1:31:3 from AA to BB. [2]
c
A circle has centre MM and passes through AA. Justify whether point BB lies inside, on, or outside this circle, giving all distances in exact form. [3]
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Solutions

3QuestionFinding Displacement Between Two PointsAssessment Practice
8 marks~12 minCriterion C
The diagram shows three line segments plotted on a coordinate grid.

Segment ABAB: A(1, 2)A(1,\,2) to B(4, 6)B(4,\,6)
Segment CDCD: C(0, 0)C(0,\,0) to D(3, 4)D(3,\,4)
Segment GHGH: G(2, 5)G(2,\,5) to H(5, 9)H(5,\,9)
a
Show that segments ABAB, CDCD, and GHGH all have the same gradient and the same length. Give lengths in exact form. [3]
b
A fourth segment connects P(k, 3)P(k,\,3) to Q(k+3, 7)Q(k+3,\,7). Find the value of kk for which QQ lies on the line through CC and DD, and hence deduce whether segment PQPQ is parallel to, or collinear with, segment CDCD. Justify your answer. [3]
c
A student claims: "Any segment with gradient 43\dfrac{4}{3} and one endpoint on the line y=43xy = \dfrac{4}{3}x must be collinear with CDCD."

Justify whether this claim is correct or incorrect, referring to the structure of a straight line. [2]
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Solutions

4QuestionPosition Vectors in Geometric DiagramsAssessment Practice
8 marks~12 minCriterion D
A surveyor is mapping a triangular plot of land with vertices at A(2,1)A(2, 1), B(8,3)B(8, 3), and C(4,7)C(4, 7), where coordinates are given in metres. A fence post is to be placed at point MM, the midpoint of side ACAC.

The diagram shows the triangular plot ABCABC with MM marked on ACAC.
a
Find the coordinates of MM and hence calculate the length BMBM, giving your answer in metres to 3 significant figures. [3]
b
The GPS device used to record each vertex has a possible error of ±2%\pm 2\% in each coordinate. Complete the table below to show the maximum and minimum possible xx-coordinates of AA and CC, and hence determine the maximum possible error in the xx-coordinate of MM. Give your answer in centimetres.

Vertex — Nominal xx (m) — Max xx (m) — Min xx (m)

AA — 22 — —

CC — 44 — — [3]
c
The surveyor claims the error in locating MM on the ground is negligible compared to the length BMBM. Advise the surveyor whether this claim is acceptable for legal boundary placement, justifying your advice with the values calculated in parts (a) and (b). [2]
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Solutions

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