RI Vector Algebra Ration Theorem Scalar and Vector Product Solns

RAFFLES INSTITUTION H2 Mathematics 9758 2023 Year 6 Term 3 Revision 6 (Summary and Tutorial) 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 ( )123,,Aa a a and ( )123,,Bbb 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 11 2 2 22 3 3 33 a b ab a b ab a b ab ±     ±= ± = ±     ±   ab and 11 22 33 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 1 222 2 123 3 a a aaa a  = = ++ a . This value is equal to the distance from O to A. We say that a is parallel to b , denoted by ab , if and only if λ=ba for some { }\0λ∈  , that is, b is a (non-zero) scalar multiple of a . • If 0λ > , then λa and a are in the same direction. • If 0λ < , then λa and a are in opposite directions. Points A and B have position vectors a and b respectively, relative to the origin O , such that λ=ba for some { }\0λ∈  . We then say that the points O, A and B are collinear.

The unit vector in the direction of a denoted by ˆa is obtained by scaling a by 1 a , thus 1ˆ =aa a . 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 ( ) 22212 2 3+ +− = so the unit vector in the direction of 1 2 2   − is 1 1 23 2   − Let a and b be non-zero and non-parallel vectors: If λµ=ab for some ,λµ ∈  , then 0.λµ= = If a b= a b stαβ++ for some , ,, ,stαβ ∈  then ,stαβ= = . Note the importance of non-parallel vectors when comparing coefficients. Suppose 1 0 0    a= and 2 0 0    b= then 62 43+=+ab ab however we cannot “compare coefficients” of vectors a and b (note that a is parallel to b) as 6 4 and 2 3≠≠ .

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 divides AB in the ratio :,λµ i.e. AP PB λ µ= . If OP= p    , then µλ λµ += + abp (MF26) Note that the Ratio Theorem is an immediate consequence of the addition of vectors. From diagram, ( )pa ba λ λµ−= − + , rearranging we have µλ λµ += + abp . Sometimes, it is easier to use ( )pa ba λ λµ−= − + like the following example. Points ,AB and P have position vectors ,ab and p respectively, relative to the origin O . Given that 2 2 5  = a and 2 6 3  =− b , find p if P lies on AB produced such that 2 5 AB AP = . Solution: Easier to find directly, ( )5