​Change The Point To The Vector

​Question:-

a. A(3,3), B(-2, 2) b. A (8, 0, -2), B(5, 3, 1)

c. A(-1,8), B(5, 9) d. A (0, 7, 5), B(11, 5, 0)

​Magnitude or length

​The length of 2-dimensional vector a =

The length of 3-dimensional vector a =

​Find the Magnitude

1.A(1 , 2 , 3) and B(3 , 3 , 4)

​2.X(1 , 3 ) and Y(1 , 4 )

​is

is

​Vector Addition

​If a=

​and b=

​,then

​and

​A)Given that a=6i -j +3k ,b=5j + k

1.a+b

​2.a-b

​1. (6i - j + 3k) + (0i + 5j + k)

=(6+0)i +(-1+5)j +(3+1)k

=6i + 4j + 4k

​2.(6i - j + 3k) - (0i + 5j + k)

=(6-0)i +(-1-5)j +(3-1)k

=6i + (-6)j + 2k

=6i - 6j + 2k

​B)Given that a=⟨-4 ,3) ,(6, 2)

1. 3a

​2.-4a + 5b

​1.3a=(-4(3) , 3(3) )

=(-12 , 9)

2.-4a + 5b =(-4(-4) ,3(-4)) + (6(5) , 2(5))

=(16 , -12) + (30 , 10)

=(46, -2)

​Algebraic Properties of Vectors

• Commutative (vector) P + Q = Q + P

• Associative (vector) (P + Q) + R = P + (Q + R)

• Additive identity There is a vector 0 such that (P + 0) = P = (0 + P) for all P

• Additive inverse For any P there is a vector -P such that P + (-P) = 0

• Distributive (vector) r(P + Q) = rP + rQ

• Distributive (scalar) (r + s) P = rP + sP

• Associative (scalar) r(sP) = (rs)P

• Multiplicative identity For the real number 1, 1P = P for each P

​Additional For Basic Knowlegde

​Unit vector

• A vector of length 1 is called a unit vector.

• In an xy -coordinate system the unit vector is denoted by i and j

• In an xyz -coordinate system the unit vector is denoted by i, j and k

​Dot Product

​If a=

​and b=

​then their dot product is:

​This also applies to 3-space vectors

​Find the Dot product

1.a=⟨4, -1⟩ ,b=⟨3, 6⟩

​2. a=i -2j +3k ,b=5i + 9k

=4(3) + 6(-1)

=6

​Properties of the Dot Product

• (Commutative Property) For any two vectors A and B, A^.B = B^.A.

• (Scalar Multiplication Property) For any two vectors A and B and any real number c, (cA)^.B = A^.(cB) = c(A^.B)

• (Distributive Property) For any 3 vectors A, B and C, A^.(B+C) = A^.B + A^.C.

​Additional information

=5(1) + 0(-2) + 9(3)

=5+0+27

=32

​1.

​2.

​Angle Between The Vectors

​The angle between two vectors a and b is found using the formula

​Find the angle between the vectors

1.a=3i + 4j – k and b=2i – j + k.

​First Step :Find the Dot Product

= (3i + 4j – k).(2i – j + k)

= (3)(2) + (4)(-1) + (-1)(1)

= 6-4-1

= 1

​Last Step:put all the values you just found into the Formula

​Cross Product

​this are for three-dimensional vectors only.

if the a=

​and b=

​then the cross product is

​Question of Cross Product

​1.Find the cross product for a=⟨1,2,3⟩ b=⟨1,2,3⟩

​2.Find the cross product for a=2i + j - k ,b=j + 2k

​i j k

​2 1 -1

​0 1 2

​=(2-(-1))i - (4-0)j + (2-0)k

=3i -4j + 2k

=⟨3 ,-4 , 2⟩