ENG1005 Week 1: Practice questions
1. Calculate a · b for
 
7
a = 1
5

and

1
b = −3 .
3
2. Calculate the length of
 
2
a = 1
7
and find a unit vector pointing in this direction.
3. Calculate the length of


4
a= 0 
−3
and find a unit vector pointing in this direction.
4. Calculate a × b for
5. Calculate a × b for


5
a= 0 
−2

 
−2
a= 3 
4
and

1
b = −3
−1
and

1
b = −5
−1

6. Calculate a · (b × c) for
 
−1
a = −5 ,
2

2
b= 3 
−5

and
 
5
c = 4
1
7. Find a vector representation of the line passing through the two points
 
 
2
−3
0
0
and
4
9
.
8. Find a vector representation of the line passing through the two points
 
 
−5
1
2
1
and
−8
1
.
1
9. Find the parametric and symmetric forms of the line
 
 
1
3
1 + s −2
1
7
10. Find the parametric and symmetric forms of the line
 
 
−1
2
 3  + s 0
0
5


−5
11. Is the point  −3  on the line
−13
 
 
1
3
x(s) = 1 + s  2 
1
−7
12. Find the minimum distance between the two lines defined by
 
 
1
1
x (t) = −2 + t −3
2
1
and
 
 
0
3




1
−2
x (s) =
+s
2
−1
when s and t are real numbers.
13. Find the minimum distance between the two lines defined by
 
 
0
5
x (t) = −2 + t  1 
−6
−2
and
 
 
−2
1
x (s) =  3  + s 2
0
5
when s and t are real numbers.
14. Find a parametric equation of the plane that passes through the points
 
 
 
1
3
−3
 1 ,
−1
−2
and
−5
0
1
15. Find a parametric equation of the plane that passes through the points
 
 
 
1
2
−1
 2 ,
0
−1
and
−1
−1
0
2
16. Given the parametric equation for the plane


 
 
1
2
−4
x(s, t) =  1  + s −2 + t −3
−5
5
6
find the algebraic equation for the plane.
17. Given the parametric equation for the plane


 
 
1
1
−2
x(s, t) =  2  + s −2 + t −3
−1
0
1
find the algebraic equation for the plane.
18. Find the normal vector for the plane defined by the equation 3x + 4y − z = 2.
19. Find the normal vector for the plane defined by the equation −2x + y + 2z = 6.
20. Consider a plane defined by the algebraic equation 3x + 4y − z = 9 and a line defined by the following
vector equation
 
 
2
−2
x(t) = −1 + t  3 
0
−1
Find the point where the line intersects the plane.
3