Geometric Proof with Vectors (DP IB Applications & Interpretation (AI): HL): Revision Note

Geometric proof with vectors

How can I prove geometric facts about a pair of vectors?

• To show that two vectors are parallel:

  • Either show that they are scalar multiples of each other

  • Or show that their vector product is equal to the zero vector

• To show that two vectors are perpendicular

  • Show that the scalar product of their vectors is equal to zero

• To show that two vectors have equal length

  • Show that their magnitudes are equal

• To show that two vectors have equal length and are parallel

  • Either show that they are equal

  • Or show that one is the negative of the other

    • They are the same lengths but in opposite directions

How can I use vectors to prove that line segments create a specific quadrilateral?

• You can represent each side of a quadrilateral using displacement vectors

• To prove that a 2D shape is a parallelogram

  • Either show that the vectors for both pairs of opposite sides are equal or opposite vectors

• To prove that a 2D shape is a rectangle

  • Show that the vectors for both pairs of opposite sides are equal or opposite vectors

  • And show that the vectors for a pair of adjacent sides are perpendicular

• To prove a 2D shape is a rhombus

  • Show that the vectors for both pairs of opposite sides are equal or opposite vectors

  • And show the vectors for a pair of adjacent sides have equal lengths

• To prove a 2D shape is a square

  • Show that the vectors for both pairs of opposite sides are equal or opposite vectors

  • And show the vectors for a pair of adjacent sides have equal lengths

  • And show that the vectors for a pair of adjacent sides are perpendicular

• To prove a 2D shape is a kite

  • Show that the vectors for both diagonals are perpendicular

  • And show that no pairs of vectors are parallel

• To prove a 2D shape is a trapezium

  • Show that the vectors for only one pair of opposite sides are parallel

How do I find midpoints and points on a line using vectors?

• If the point $\mathrm{M}$ is the midpoint of the line segment $\mathrm{AB}$ then

  • $\overrightarrow{\mathrm{AM}}=\frac{1}{2}\overrightarrow{\mathrm{AB}}$

• More generally, if the point $\mathrm{X}$ divides a line segment $\mathrm{AB}$ into the ratio $p:q$ then

  •  $\overrightarrow{\mathrm{AX}}=\frac{p}{p+q}\overrightarrow{\mathrm{AB}}$

  •  $\overrightarrow{\mathrm{XB}}=\frac{q}{p+q}\overrightarrow{\mathrm{AB}}$

• The position vector of the midpoint of points $\mathrm{A}$ and $\mathrm{B}$ is $\frac{1}{2}\left(\mathit{a}+\mathit{b}\right)$

  • $\mathit{a}$ is the position vector of $\mathrm{A}$

  • $\mathit{b}$ is the position vector of $\mathrm{B}$

How can vectors be used to prove that three points are collinear?

• Three points are collinear if they all lie on the same line

• To show that the points $\mathrm{A}$, $\mathrm{B}$ and $\mathrm{C}$ are collinear

  • Show that any two of $\overrightarrow{\mathrm{AB}}$, $\overrightarrow{\mathrm{AC}}$ and $\overrightarrow{\mathrm{BC}}$ are parallel