Geometric Proof with Vectors (DP IB Applications & Interpretation (AI)): 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 kite

  • 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 is the midpoint of the line segment then

  • 

• More generally, if the point divides a line segment into the ratio then

  •  

  •  

• The position vector of the midpoint of points and is

  • is the position vector of

  • is the position vector of

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 , and are collinear

  • Show that any two of , and are parallel