Given u 372 v 621 and w 238 verify the following

12. Given
⃗ u =(− 3,7,2 )
,
⃗ v =( 6,2,1 )
, and
⃗ w =(− 2,3,8 )
, verify the following arithmetic properties
of vectors. (12 marks: 4 marks each)
w

The cross product is
⃗ u× ( ⃗ v + ⃗ w ) = ( 53,35, − 43 ) .
Right side:
⃗ u× ⃗ v + ⃗ u× ⃗ w
¿
(
− 3,7,2
)
×
(
6,2,1
)
+
(
− 3,7,2
)
×
(
− 2,3,8
)
¿ ( 3,15, − 48 ) +( 50,20,5 )
¿ ( 53,35, − 43 )
The cross product is
⃗ u× ⃗ v + ⃗ u× ⃗ w = ( 53,35, − 43 ) .
Since
⃗ u× ( ⃗ v + ⃗ w ) =( 53,35, − 43 )
and
⃗ u× ⃗ v + ⃗ u× ⃗ w = ( 53,35, − 43 )
, we can conclude that
⃗ u× ( ⃗ v + ⃗ w ) = ⃗ u× ⃗ v + ⃗ u× ⃗ w
and the distributive law holds true for cross products.

b)
⃗ u∙ ( ⃗ v + ⃗ w ) = ⃗ u∙ ⃗ v + ⃗ u∙ ⃗ w
Left side:
⃗ u∙ ( ⃗ v + ⃗ w )
¿
(
− 3,7,2
)
∙ [
(
6,2,1
)
+
(
− 2,3,8
)
]
¿ ( − 3,7,2 ) ∙ ( 4,5,9 )
¿
(
− 3
) (
4
)
+
(
7
) (
5
)
+( 2 )( 9 )
¿ − 12 + 35 + 18
¿ 41
Right side:
⃗ u∙ ⃗ v + ⃗ u∙ ⃗ w
¿
(
− 3,7,2
)
∙
(
6,2,1
)
+
(
− 3,7,2
)
∙ (− 2,3,8 )
¿ ( − 3 ) ( 6 ) + ( 7 ) ( 2 ) + ( 2 ) ( 1 ) + ( − 3 ) ( − 2 ) + ( 7 ) ( 3 ) +( 2 )( 8 )
¿ − 18 + 14 + 2 + 6 + 21 + 16
¿ 41
Therefore,
⃗ u∙
(
⃗ v + ⃗ w
)
= ⃗ u∙ ⃗ v + ⃗ u∙ ⃗ w
and the distributive law applies to the dot product of two vectors.
c)
( ⃗ u + ⃗ v ) ∙ ( ⃗ u + ⃗ v ) =
|
⃗ u
|
2
+
|
⃗ v
|
2
+ 2 ( ⃗ u∙ ⃗ v )
Left side:
( ⃗ u + ⃗ v ) ∙ ( ⃗ u + ⃗ v )
¿
[
(
− 3,7,2
)
+
(
6,2,1
)
]
∙
[
(
− 3,7,2
)
+
(
6,2,1
)
]
¿ ( 3,9,3 ) ∙ ( 3,9,3 )

¿ ( 3 ) ( 3 ) + ( 9 ) ( 9 ) +( 3 )( 3 )
¿ 9 + 81 + 9
¿ 99
Right side:
|
⃗ u
|
2
+
|
⃗ v
|
2
+ 2 ( ⃗ u∙ ⃗ v )
¿
( √
(
− 3
)
2
+
(
7
)
2
+
(
2
)
2
) 2
+
( √
(
6
)
2
+
(
2
)
2
+
(
1
)
2
) 2
+ 2
(
− 3,7,2
)
∙ ( 6,2,1 )
¿ 62 + 41 + 2
(
( − 3 ) ( 6 ) + ( 7 ) ( 2 ) + ( 2 ) ( 1 )
)
¿ 103 + 2 (− 2 )
¿ 99
Therefore ,
(
⃗ u + ⃗ v
)
∙
(
⃗ u + ⃗ v
)
=
|
⃗ u
| 2
+
|
⃗ v
| 2
+ 2 ( ⃗ u∙ ⃗ v )
.
5/5 marks
13. Describe how the dot product can be used to determine whether two vectors are perpendicular. Create
a question with vectors in 3-space to illustrate this property. Be sure to solve the question as well. (5 marks)
is:
z
2

⃗ u∙ ⃗ v = x
1
x
2
+ y
1
y
2
+ z
1
z
2
= 0
For example, find the dot product of the following pair of vectors. Determine whether vectors
⃗ u =( 1,9,3 )
and
⃗ v =( 6,2, − 8 )
are perpendicular:
Find the dot product using the Cartesian formula:
⃗ p∙ ⃗ q = ( 1 ) ( 6 ) + ( 9 ) ( 2 ) +( 3 )(− 8 )
¿ 6 + 18 − 24
¿ 0
The dot product is 0, so the vectors
⃗ p =( 1,9,3 )
and
⃗ q =( 6,2, − 8 )
are perpendicular.
Task 4: Application questions
7/7 marks