Geometrical Proofs Using Vectors for Problem Solving
Explore how vectors can be utilized to deduce geometrical results through proofs and solve problems related to physical quantities like force and velocity. Witness step-by-step demonstrations in triangles showcasing vector addition, subtraction, and scalar multiples for in-depth understanding.
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20 March 2025 Geometrical proofs LO: Use vector addition, subtraction and scalar multiples to deduce some geometrical results. www.mathssupport.org www.mathssupport.org
Using vectors to solve problems We can use vectors to solve many problems involving physical quantities such as force and velocity. We can also use vectors to prove geometric results. For example, suppose we have a triangle ABC as follows: The line PQ is such that P is the mid-point of AB and Q is the mid-point of AC. B P Use vectors to show that PQ is parallel to BC and that the length of BC is double the length of PQ. C A Q www.mathssupport.org www.mathssupport.org
Using vectors to solve problems Let s call vector a and vector b. AP AQ a b = + PQ B = b a P = 2 +2 a b BC a = 2 2 b a C b A Q =2( ) b a Therefore, = 2 BC PQ We can conclude from this that PQ is parallel to BC and that the length of BC is double the length of PQ. www.mathssupport.org www.mathssupport.org
Using vectors to solve problems In triangle OXY, A, B and C are the mid points of OX, OY and XY respectively. (a) Find expressions for ??,??, ??,?? ??? ?? in terms of x and y ?? = and ?? = ? ?? = ? 1 2? 1 2? 1 2?? = 1 2?? ?? + ?? = ? + ? ?? + ?? = ?? +1 X x = ?? = A C = ? ? ?? = Y O B y 2?? ?? = =1 2? +1 2? =1 = ? +1 2? + ? 2? ? = 1 2? + ? ?? = ?? www.mathssupport.org www.mathssupport.org
Using vectors to solve problems In triangle OXY, A, B and C are the mid points of OX, OY and XY respectively. and ?? = ? ?? = ? (b) Find an expression for ?? in terms of x and y ?? + ?? = 1 2? +1 2? ?? = X =1 x 2? ? A C (c) What is the relationship between the line XY and the line AB? Y O B y 1 2? ? ?? = ? ? ?? = Then the line XY is double the length of the line AB and the same direction as AB Therefore the line XY is parallel to the line AB www.mathssupport.org www.mathssupport.org
Using vectors to solve problems In triangle OXY, A, B and C are the mid points of OX, OY and XY respectively. and ?? = ? ?? = ? (d) P is a point such that ?? = ?? +2 3?? . Find ?? ?? +2 3?? +2 3 ? +1 ?? = X x ? ?? = ?? + ?? A C ? +2 2? ?? = 3 Y O B y 3? =1 =1 3? +1 = ? 2 3?+1 3? + ? 3? 1 3? + ? ?? = www.mathssupport.org www.mathssupport.org
Using vectors to solve problems In triangle OXY, A, B and C are the mid points of OX, OY and XY respectively. (e) What you can conclude about the position of P? and ?? = ? ?? = ? ?? +2 3?? +2 3 ? +1 ?? = X x ? ?? = ?? + ?? A C ? +2 2? ?? = 3 Y O B y 3? =1 =1 3? +1 = ? 2 3?+1 3? + ? 3? 1 3? + ? ?? = www.mathssupport.org www.mathssupport.org
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