Assignment 1
A = (2, 0, −2)
B = (1, −1, 1)
C = (4, 2, 3)
→ BA
→ and BC
→ .
a) Find vectors AC,
→
AC = ⟨4 − 2, 2 − 0, 3 − (−2)⟩ = ⟨2, 2, 5⟩
→
BA = ⟨2 − 1, 0 − (−1), −2 − 1⟩ = ⟨1, 1, −3⟩
→
BC = ⟨4 − 1, 2 − (−1), 3 − 1⟩ = ⟨3, 3, 2⟩
b) Which pairs of these vectors are orthogonal to one another?
→ ⋅ BA
→ = 2 ⋅ 1 + 2 ⋅ 1 + 5 ⋅ (−3) = −11
AC
→
→
→
→
AC ⋅ BC = 2 ⋅ 3 + 2 ⋅ 3 + 5 ⋅ 2 = 22
BA ⋅ BC = 1 ⋅ 3 + 1 ⋅ 3 + (−3) ⋅ 2 = 0
→
→ are orthogonal to one another.
∴ BA, BC
→ ⋅ BA
→ and the angle between AC
→ and BA
→
c) Find AC
→
→
AC ⋅ BA = 2 ⋅ 1 + 2 ⋅ 1 + 5 ⋅ (−3) = −11
→
|AC| = √ 22 + 22 + 52 = √ 33
→
2
|BA| = √ 1
−1
θ = cos
2
+ 1
→
2
+ (−3)
= √ 11
→
AC ⋅ BA
(
→
−1
) = cos
→
−11
∘
(
) = 125.26
√ 33 ⋅ √ 11
|AC||BA|
→ and BA
→ .
d) Find a unit vector that is orthogonal to both AC
→
→
i
⎡^
^
j
^
k
⎤
2
2
5
1
1
−3
AC × BA =
⎣
⎦
^
= [(2)(−3) − (5)(1)] ^
i − [2(−3) − (5)(1)] ^
j + [(2)(1) − (2)(1)] k
^
= −11^
i + 11^
j + 0k
= ⟨−11, 11, 0⟩
→
→
|AC × BA| = √ −11
→
^ =
u
2
→
2
+ 0
= √ 242
⟨−11, 11, 0⟩
AC × BA
→
→
|AC × BA|
2
+ 11
=
−11
= ⟨
√ 242
11
,
√ 242
, 0⟩
√ 242