MAT-NYC
Lab on 3-dimensional coordinate system
Recall:
1. Give the coordinates of all the vertices of the following parallelepipeds.
a)
b)
2. Plot the following points in R3 :
a) (0, 0, 3)
b) (3, 0, 3)
c) (3, 4, 2)
d) (1, 3, 1)
e) (3, 4, 5)
f) (3, 4, 5)
c
MAT-NYC
LAB #8
Intersections and distances
1. Determine whether the lines L1 and L2 coincide, are parallel, intersect, or are skew. If
they do intersect, nd their point of intersection and their angle of intersection.
a) L1 : x = 3+2t, y = 4 6t, z = 2+4t
MAT-NYC
LAB #7
Lines and Planes
1. Find both symmetric and parametric equations for the line
a) through P (2, 1, 3) and perpendicular to the plane 2x + y = 1.
b) through P (1, 1, 1) and perpendicular to both the lines
L1 : x = 2 + t, y = t, z = 1 2t and L
MAT-NYC
LAB #4
Subspaces - Linear Spans - Solution Spaces
1. Rewrite the following as a vector equation.
2x1 + 6x2 + x3
a) The linear system
x 1 + 5 x2
2x2 3x3
b) The matrix equation
7x4 = 1
1
x4 = 0 .
2
=5
2
4 2 5
9 6
=
7
1 8 3 1
0
2. Rewrite the vect
MAT-NYC
LAB #3
Linear Systems
2x1 + 6x2 + x3 7x4 = 1
1
1. Given the m n -system
x1 + 5x2
x4 = 0
2
2x 3x
=5
2
3
a) What are m and n?
b) Write the system in matrix form.
2. Which of the following matrices are in
i) row-echelon form?
ii) reduced row-echelo
MAT-NYC
LAB #2
Basic Matrix Algebra
ab
cd
1. Find a, b, c and d if
=
c 3d d
.
2a + d a + b
2. Compute, where possible,
a) 3A 2B
b) 4AT 3C
A=
3 1 2
,C=
0
14
2
1
,B=
0 1
3. Solve 5X
c) B + 2DT
101
010
3 1
2
0
011
111
=3 X +2
d) (A + C )T ,
where
1 3
and D
MAT-NYC
LAB #1
Vectors - Norm and dot product
1. Given u and v (see sketch), draw each of the following vectors on separate sketches
using geometric constructions only.
a) u + v
b) u v
d) u + 2v
c) 2u + v
e) the vector w such that 2u + 3v w = 0
f) the vec
MAT-NYC
LAB #8
Intersections and distances
1. Determine whether the lines L1 and L2 coincide, are parallel, intersect, or are skew. If
they do intersect, nd their point of intersection and their angle of intersection.
a) L1 : x = 3+2t, y = 4 6t, z = 2+4t