When three or more points lie on a single straight line, these points are said to be collinear.

Geometric proofs with vectors

Maths

In the prior unit, pupils described a translation using column vector notation. Pupils' understanding of vectors is further developed in this unit, with pupils exploring column vector notation and arithmetic procedures.

Vectors

Three points that lie along the same straight line are said to be {{}}.

$$\overrightarrow{OD}=2\overrightarrow{EF}$$

$$\overrightarrow{OD}=2\overrightarrow{FE}$$

$$\overrightarrow{OD}=\frac{1}{2}\overrightarrow{FE}$$

$$\overrightarrow{OF}=k\overrightarrow{ED}$$

$$\overrightarrow{OF}\neq k\overrightarrow{ED}$$

Select the correct statements to show that $$ODEF$$ is a trapezium.

$$\overrightarrow{OF}=\overrightarrow{AE}$$

$$\overrightarrow{OF}=2\overrightarrow{AE}$$

$$2\overrightarrow{OF}=\overrightarrow{AE}$$

$$\overrightarrow{OA}=\overrightarrow{FE}$$

$$\overrightarrow{OA}\neq k\overrightarrow{FE}$$

Select the correct statements to show that $$OAEF$$ is not a parallelogram.

$$ABCD$$ is a kite. $$W,X,Y,Z$$ are midpoints of the lines they lie on. $$\overrightarrow{XW} = k(\mathbf b - \mathbf a)$$ where $$k= $$ {{}}

$$ABCD$$ is a kite. $$W,X,Y,Z$$ are midpoints of the lines they lie on. Find $$\overrightarrow{ZW}$$.

$$-2 \mathbf c -4 \mathbf b + 2\mathbf a$$

$$ 2\mathbf c + 4\mathbf b - 2\mathbf a$$

$$ABCD$$ is a kite. $$W,X,Y,Z$$ are midpoints of the lines they lie on. Find $$\overrightarrow{XZ}$$.

Work out the magnitude of the vector $${4 \choose -3}$$.

Which of the following vectors are parallel to $$2\mathbf{a}-3\mathbf{b}$$?

Which of the following vectors are parallel to $${8\choose -12}$$?

$$\frac{1}{5}\overrightarrow{AB}=\overrightarrow{AX}$$    

$$\frac{3}{11}\overrightarrow{AB}=\overrightarrow{AX}$$  

$$\frac{1}{2}\overrightarrow{AB}=\overrightarrow{AX}$$    

$$\frac{3}{4}\overrightarrow{AB}=\overrightarrow{AX}$$   

X is a point on the line AB. Match each vector statement to the correct ratio.

$${x \choose -2} + {12 \choose 7} = {25 \choose 5} $$

$${x\choose -7} + {-7 \choose -7} = {5 \choose -14} $$

$$3{3\choose 4} + {3 \choose -4} = {12 \choose x} $$

Match each value of $$x$$ to the vector equation that it satisfies.

$$\frac{1}{2}\overrightarrow{AB}=\overrightarrow{MN}$$

$$\overrightarrow{MN}=2\overrightarrow{AB}$$

$$\frac{1}{3}\overrightarrow{AB}=\overrightarrow{MN}$$

$$2\overrightarrow{MN}=\overrightarrow{AB}$$

$$\overrightarrow{AB}=\frac{1}{2}\overrightarrow{MN}$$

OACB is a parallelogram. M is the midpoint of OA and N is the midpoint of OB. $$\overrightarrow{OM}=2\mathbf{a}$$ and $$\overrightarrow{ON}=2\mathbf{b}$$.

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