1. 3D unit vectors Linear Combinations

2. |a| = (r2 + s2 + t2) Magnitude of a 3D Vector (General) z a y o x t s
“the magnitude is the square root
of the sum of the squares of
the 3 components.” x
[Pythagoras in 3D]

3. i is the unit vector in the x-direction
Unit Vectors (1)
i is the unit vector in the x-direction z j y k
j is the unit vector in the y-direction i x
k is the unit vector in the z-direction
All vectors can be expressed as a linear combination of these 3 vectors

4. a = i - 2j - 6k Unit Vectors (2) z y a x All 3D Vectors can be
expressed in this form

5. a = i - 2j - 6k |a| = (12 + (-2)2 + (-6)2) |a| = (1 + 4 + 36) = 41
3D Displacement Vectors x y z
a = i - 2j - 6k
|a| = (12 + (-2)2 + (-6)2) a
|a| = ( ) = 41
|a| = 6.4

6. Linear Combinations
3a + 6b
2a - b

7. + = (6i + 4j) + (i - 5j) = 7i - j Linear Combination Example 64 1 7 -5
A car drives from A to B with displacement = 6i + 4j
Then he drives from B to C with displacement = i - 5j
6i + 4j B A C
i - 5j 1 -5 64 = 7 -1
The resultant displacement is from A to C
(6i + 4j) + (i - 5j) = 7i - j
The magnitude of the displacement
= (72 + (-1)2) = 50 = 7.1m (1 d.p.)

8. 3D Vectors
Follow all the same rules of 2D Vectors!

9. 3D Displacement Vectors - addingx y z
a + b
a = i - 2j - 6k
b = 3i + 4j + 11k a b
a + b = (i - 2j - 6k) + (3i + 4j + 11k)
a + b = (4i + 2j + 5k)

10. Given: a = 3i + 4j and b = i - 3j Vector Problem
Find x and y, if xa + yb = 11i + 6j
xa + yb = x(3i + 4j) + y(i - 3j)
= 3xi + 4xj + yi - 3yj
= (3x+y)i + (4x-3y)j
= 11i + 6j
Therefore: 3x + y = 11 (i parts)
x3 9x + 3y = 33
13x = 39 +
and: 4x - 3y = 6 (j parts)
4x - 3y = 6
3x + y = 11
Substitute
9 + y = 11
x = 3
y = 2