T2W10_Vectors+I_Summary

RAFFLES INSTITUTION H2 Mathematics 9758 2025 Year 6 Term 2 Revision (Summary) Topic: Vectors 1 (Vector Algebra, Ratio Theorem, Scalar and Vector Product) Summary for Vectors 1 Vector Algebra With reference to an origin O(0, 0, 0), given points  1 2 3, ,A a a a and  1 2 3, ,B b b b, we have the corresponding (position vectors) expressed in column form 1 2 3 a a a       a and 1 2 3 b b b       b . 1 1 1 1 2 2 2 2 3 3 3 3 a b a b a b a b a b a b a b                                and 1 1 2 2 3 3 a a ka k k a ka a ka                  with k a real number. The magnitude (or modulus) of a vector, a, is the non-negative number 1 2 2 2 2 1 2 3 3 a a a a a a          a . This value is equal to the distance from O to A. We say that ais parallel to b, denoted by a b, if and only if b a for some  \ 0, that is, b is a (non-zero) scalar multiple of a.  If 0 , then a and a are in the same direction.  If 0 , then a and a are in opposite directions. Points A and B have position vectors a and b respectively, relative to the origin O, such that b a for some  \ 0. We then say that the points O, A and B are collinear.

Summary on Vectors 1 Page 2 of 6 The unit vector in the direction of a denoted by ˆa is obtained by scaling a by 1 a , thus 1ˆa aa . The vectors 1 2 2       and 2 4 4       are parallel since 2 4 4       is a scalar multiple of 1 2 2       (k = –2) but are in opposite directions since k < 0. The points 2, 4, 4 , 0,0,0 and (1, 2, 2)   are also said to be collinear. The magnitude of 1 2 2       is   22 21 2 2 3    so the unit vector in the direction of 1 2 2       is 11 23 2       Let a and b be non-zero and non-parallel vectors:  If  a b for some ,  , then 0.    If s t  a b = a b for some , , , ,s t   then ,s t   . Note the importance of non-parallel vectors when comparing coefficients. Suppose 1 0 0       a = and 2 0 0       b = then 6 2 4 3  a b a b however we cannot “compare coefficients” of vectors a and b (note that a is parallel to b) as 6 4 and 2 3  .

Summary on Vectors 1 Page 3 of 6 Ratio Theorem Consider a triangle OAB with OAa  and OBb  . So a and b are non-zero and non-parallel vectors. Let P be a point which dividesAB in the ratio : ,  i.e. AP PB   . If OPp  , then       a bp (MF27) Note that the Ratio Theorem is an immediate consequence of the addition of vectors. From diagram,      p a b a , rearranging we have       a bp . Sometimes, it is easier to use      p a b a like the following example. Points , A B and Phave position vectors , a b and p respectively, relative to the origin O. Given that 2 2 5  