Vectors Flashcards
Two types of measurable quantity
Scalar
Vector
Scalar
A quantity with a magnitude (size)
Vector
A quantity with a magnitude and direction
When multiplying a vector by a scalar, what stays the same and what changes
The direction remains the same, the magnitude changes
Component form of vector
( x y z )
given the points P (3,4), Q(5,2) and O (0,0), find the vector joining them
(2, -2)
PQ = ?
q - p
How is magnitude written
|u| or |AB|
How is magnitude of a vector calculated
Using Pythagoras
|PQ| = ?
/a^2 + b^2
What is a unit vector
A parallel vector to the original vector with a magnitude of 1
Unit vector magnitude
/l^2 + m^2 + n^2 = 1
l^2 + m^2 + n^2 = 1
How to find unit vector of original vector
Divide original vector by its magnitude
How many unit vectors does a vector have
Two, +ve and -ve
Method for unit vectors
Find magnitude of original vector
Divide vector by its length
Find the components of the unit vector, u, parallel to vector v if v = (3 4 )
( 3 / 5, 4 / 5 )
Find two unit vectors of the vector a = ( 12, 3, 4 )
+ - ( 12/13 , 3/13, 4/13)
What to use to find a component of a unit vector
/ l^2 + m^2 + n^2 = 1
u = (a, 1/3, 1/3) is a unit vector.
Find the two values of a
+ - /7 / 3
Method for proving unit vector
Use l^2 + m^2 + n^2 = 1
Statement to prove
Is (2/3 1/3 -2/3 ) a unit vector
Yes since l^2 + m^2 + n^2 = 1
Orgin coordinates
0, 0
0, 0
Origin
unit vector i
( 1 0 0)
( 1 0 0)
Unit vector i
Unit vector j
( 0 1 0)
( 0 1 0)
Unit vector j
Unit vector k
( 0 0 1)
( 0 0 1)
Unit vector k
i, j, k =
i = x
j = y
k = z
Express 4i - 8j + 3k in component form
( 4 -8 3)
Express 7j - 2k in component form
( 0 7 -2)
Express (2 5 -1) in the form of i, j, and k
2i + 5j - k
Express (-2 0 5) in the form of i, j, and k
-2i + 5k
The vector, a, has the components (-3 0 4).
Find the unit vector parallel to a in the form i, j, k.
(-3/5 0 4/5)
= -3/5i + 4/5k
Parallel vectors relationship
if vector u = ( x y ), then ku = (kx ky)
Method for proving if 2 vectors are parallel
- Take out highest common factor in each vector
- Check for same vector
- Express vector in terms of other
Prove u = (2 3) and v (6 9) are parallel
3u = v
Show a = (6 -2) and b = (21 -7) are parallel
a = 2/7b
show that p = (4 -10) and q = ( -6 15) are parallel
p = 2/-3q
are c = (12 9) and d = (16 8) parallel?
No, (4 3) ≠ ( 2 1)
Proving collinearity of vectors method
- Find vector of first 2 coordinates
- Find vector of last 2 coordinates
- Statement
Prove the points A (2, 4) B (8, 6) and C (11, 7) are collinear with vectors
2BC = AB so parallel and share common point B
Prove the points A (0, 1, 2) B (1, 3, -1) and C (3, 7, -7) are collinear with vectors
Parallel since 2BC = AB
and share common point B
Prove the points A (2, -3, 4) B (8, 3, 1) and C (12, 7, -1) are collinear with vectors
AB = 2/3 BC so parallel and common point B
The points A (-8, a, -12) B (2, -5, 3) and C (6, -15, b) are collinear. Find a and b
a = 20
b = 9
Section formula
na + mb/ ( m + n )
remember,
A P B
m n
P divides AB internally in the ratio 3:2. How do you write this
AP:PB = 3:2
P divides AB externally in the ratio 3:2. How do you write this
AP:PB = 3:-2 OR -3:2
Section formula method
- Identify ratio and points
- Use cross method to match up points and ratio
- Sub into formula
- Solve
AP:PB = 2:5
Find P, given that A is (3, 4) and B is (17, 11)
(7, 6)
P divides AB in the ratio 3:2
Find P given that A is (2, -3, 4) and B is (12, 7, -1)
P is (8, 3, 1)
Triangle ABC has the vertices A (3, 0, 6), B (0, 3, -3) and C (1, 0, -4). P divides AB in the ratio 1:2, Q is the midpoint of AC and R divides BC externally in the ratio 2:1.
Find the coordinates of P, Q and R.
P = (2, 1,3)
Q = (2, 0, 1)
R = (2, -3, -5)
Find the coordinates of the point dividing joining E (5, 2, 1) and F (9, 10, 13). It divides externally in the ratio 1:3
(3, -2, -5
VABCO is a pyramid.
AB = 8i + 2j + 2k
AD = -2i + 10j - 2k
AV = i + 7j + 7k
Express CV in component form
( -5 -5 7)
EABCD is a pyramid
EA = -7i - 13j - 11k
AB = 6i + 6j - 6k
AD = 8i - 4j + 4k
K divides BC in the ratio 1:3. Find EK
EK = i- 8j - 16k
RATU, VWXY is a cuboid
Express VT in terms of f, g, and h
- f - h + g
Scalar product equation (3 forms)
a.b = |a| |b| cos0
Where 0 is the angle between a and b (underlined)
a.b = a1b1 + a2b2 + a3b3
Where a = (a1 a2 a3) and b = (b1 b2 b3)
Cos0 = a.b / |a||b|
3 things to remember with scalar product
Vectors must be pointing away from the angle
The scalar product is a scalar, not a vector!
The an and b (underlined) are perpendicular, then a.b = 0
Find scalar product:
6
Find scalar product
1/2
Find scalar product:
0
Find the scalar product
-1
Find the scalar product:
-6
Find the scalar product:
-10 root 3
Find scalar of a (1 2 5) and b (-1 1 -2)
-9
Given P (1, 5, 8), Q (-2, 1, 3) and R (1, -6, 0), if PQ represents u and QR represents v, find u.v.
u.v = 34
S is (1, 2, -3), T is (11, -3, 7)
U divides ST in the ratio 3:2. Find:
Coordinate of U
Value of SU.UT
a) (7, -1, 3)
b) 54
Angle between two vectors formula
Cos0 = a.b / |a||b|
a = i + 2j + 2k and b = 2i + 3j + 6k.
Calculate the angle between a and b (underlined a and b in all text)
17.8 degrees
Show that an and b are perpendicular if a = (2 3 -6) and b = (-3 2 0). a and b underlined
a.b = 0 since perp.
Cos90 = 0
Vectors a = 2i - 5j + k and b = pi - 2j + 3k are perpendicular. (a and b underlined)
Find the value of p
p = -7
R is (4, 4, 1) S is (3, 2, 0), T is (2, 0, 1). Calculate the size of the angle RST
131.8
scalar product if 0 < theta < 90
+ve
Scalar product if theta = 90
0
scalar product if 90 < theta < 180
-ve
a.(b + c) expanded
a.b + a.c
(a + b) . (c + d) expanded
a.c + a.d + b.c + b.d
unit vector.unit vector
0
let a = 2i - k, b = i + 2j + k, c = -j + k.
evaluate a.b + a.c
make a deduction and about a and b + c
a.b + a.c = 0
a is perpendicular to b + c
26
16
-36
