Grade 10 · Core Mathematics · Lesson 18 of 21

Determining Vector Midpoints and Endpoint Coordinates

A Grade 10 Core Mathematics CBE lesson from the sub-strand Vectors I. In it, learners calculate the midpoint of a line segment using vector addition and position vector formulas.

Strand Measurements and GeometrySub-strand Vectors ILesson code 10CM-28-L1840 minutes · 3 slides

KICD learning outcome (j)

Determine the midpoint of a vector in different situations.

By the end of this lesson, learners should be able to

Calculate the midpoint of a line segment using vector addition and position vector formulas.
Determine the coordinates of an endpoint when given a midpoint and one endpoint.
Apply vector midpoints and magnitudes to calculate lengths of line segments in triangles.

Key vocabulary

Midpoint vector: The position vector identifying the exact centre point between two directed spatial positions.; We calculated the midpoint vector by finding half the sum of position vectors $a$ and $b$ .
Column vector: A vertical array of numbers representing horizontal and vertical displacements or position coordinates.; The column vector $a = (\binom{- 4}{6})$ shows negative horizontal position.
Magnitude: The scalar length of a vector computed using the Pythagorean formula on components.; The magnitude of vector $\vec{D E} = (\binom{6}{0})$ is $\sqrt{6^{2} + 0^{2}} = 6$ units.

Lesson preview

Slide 1 · Determining Vector Midpoints via Displacement and Position VectorsModel · 12 min

Figure 1: A graph showing the position vectors for endpoints A and B alongside the calculation of midpoint M using vector addition.

To determine the midpoint $M$ of a line segment connecting points $A$ and $B$ , vector methods offer two clear approaches. First, we compute displacement $\vec{A B} = b - a$ , take half of this displacement, and add it to position vector $\vec{O A}$ to get $\vec{O M} = \vec{O A} + \frac{1}{2} \vec{A B}$ . Alternatively, using vector addition, the midpoint position vector simplifies directly to $m = \frac{1}{2} (a + b)$ . Both vector calculations yield identical Cartesian coordinates for the midpoint. Use Figure 1 to see how adding half of the displacement vector to the position vector finds the midpoint.

Vector displacement $\vec{A B}$ is obtained by subtracting position vector $\vec{O A}$ from position vector $\vec{O B}$ .
Adding half of displacement vector $\vec{A B}$ to $\vec{O A}$ gives position vector $\vec{O M}$ of midpoint $M$ .
The position vector formula $m = \frac{1}{2} (a + b)$ directly yields the midpoint coordinates.
Converting between position vectors and Cartesian coordinates ensures exact solutions in coordinate geometry.

Worked example. Write position vectors:

\vec{O A} = (\binom{3}{2})

and

\vec{O B} = (\binom{9}{10})

. Find vector

\vec{A B}

\vec{A B} = \vec{O B} - \vec{O A} = (\binom{9 - 3}{10 - 2}) = (\binom{6}{8})

. Calculate half displacement:

\frac{1}{2} \vec{A B} = \frac{1}{2} (\binom{6}{8}) = (\binom{3}{4})

. Find position vector

\vec{O M}

\vec{O M} = \vec{O A} + \frac{1}{2} \vec{A B} = (\binom{3}{2}) + (\binom{3}{4}) = (\binom{6}{6})

. Answer: The position vector is $\vec{O M} = (\binom{6}{6})$ and the coordinates of $M$ are $(6, 6)$ ..

Check question: Points $C (1, 5)$ and $D (7, 11)$ are given. Find the coordinates of midpoint $N$ using $n = \frac{1}{2} (c + d)$ .

$(4, 8)$
$(8, 16)$
$(3, 3)$
$(6, 6)$

Show answer

$(4, 8)$ . Using the formula, $n = \frac{1}{2} ((\binom{1}{5}) + (\binom{7}{11})) = \frac{1}{2} (\binom{8}{16}) = (\binom{4}{8})$ , which gives the Cartesian coordinates $(4, 8)$ .

Slide 2 · Calculating Midpoints with Negative Coordinates and Position VectorsPractice · 14 min

Figure 2: A coordinate plane displaying a line segment with negative and positive endpoint coordinates to demonstrate finding the midpoint.

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All 3 slides with diagrams and class activities
Teacher notes, SNE adaptations and assessment checks
Homework and the other 20 lessons in this sub-strand

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Determining Vector Midpoints and Endpoint Coordinates

By the end of this lesson, learners should be able to

Key vocabulary

Lesson preview

Teach the full lesson on the projector

All 21 lessons in Vectors I