13010004_Unit 7 MCV4U Grade 12 Calculus and vectors...

Unit 7
MCV4U
Grade 12
Calculus and Vectors
Task 1:
1. Determine the angle between each of the following pairs of vectors.
(a).
⃗ u = ( 2, − 4 ) ∧ ⃗ v =( 1, − 3 )

θ = 8 °
The angle between
⃗ u ∧ ⃗ v
is 8°.
(b).
⃗ p = ( − 1,4,5 ) ∧ ⃗ q =( 3, − 1,3 )
⃗
p∙
⃗
q =(− 1,4,5 ) ∙ ( 3, − 1,3 )
¿ ( − 1 ) ( 3 ) + ( 4 ) ( − 1 ) +( 5 )( 3 )
¿ − 3 − 4 + 15
¿ − 7 + 15
¿ 8
|
⃗ p
|
=
√
( − 1 )
2
+( 4 )
2
+( 5 )
2
=
√
1 + 16 + 25 =
√
42
|
⃗ q
|
=
√
( 3 )
2
+(− 1 )
2
+( 3 )
2
=
√
9 + 1 + 9 =
√
19
cos θ =
⃗ p∙ ⃗ q
|
⃗ p
||
⃗ q
|
42
√
¿
¿
19
√
¿
¿
¿
cos θ =
8
¿
cos θ =
8
√
798
cos θ =( 0.2832 )
θ = cos
− 1
( 0.2832 )
θ = 74 °
The angle between
⃗ p ∧ ⃗ q
is 74°.
2. Find the slope of the vector that is perpendicular to scalar equation

3. Write an alternate vector equation for the following line. Change both the point
and the direction vector:
4. Determine whether the angle between each of the following pairs of vectors is
acute, obtuse, or neither.
(a).
⃗ a =
(
10, − 4,1
)
⃗
b =( 3, − 2,2 )

The dot product is positive, so the angle between the vectors is acute.
(b).
⃗ p = ( 0,4, − 3 ) ⃗ q =( 7, − 2,1 )
⃗
p∙
⃗
q =( 0,4, − 3 ) ∙ ( 7, − 2,1 )
¿ ( 0 ) ( 7 ) + ( 4 ) ( − 2 ) +(− 3 )( 1 )
¿ 0 − 8 − 3
¿ − 11
The dot product is negative, so the angle between the vectors is obtuse.
5. Given the vector equation of a line in 2-space,
( x, y ) = ( 3,2 ) + t ( 2,4 ) .
Write a
scalar equation for the line.
6. Write a vector equation for the line that passes through the point
P (− 1,0,3
and is parallel to the y -axis.
)

( x, y ,z ) = ( − 1,0,3 ) + t ( 0,1,0 )
.