Dubai
Year 11
Engineering Mathematics
Revision Booklet
(End of Term 1 Exam)
1
LC 2 Chapter 8.1 Geometric Vectors
1.
Choose the correct answer
(i) A quantity with only magnitude is called a _____.
(a) Matrix
(iii) Resultant
(b) Vector
(iv) Scalar
(ii)The sum of two or more vectors is called _____.
(c) The magnitude
(iii) A scalar
(d) The resultant
(iv) The zero vector
(iii)If both the initial and terminal points of a vector are at the origin, the vector is __________.
(i)Positive
(iii) The zero vector
(ii)linearly independent
(iv) Negative
(iv)The magnitude of the zero vector is _____.
(i) 0
(iii) – 0
(ii) 1
(iv) – 1
(v)A vector that has a magnitude of one is called a _____.
(i) Scalar
(iii) one vector
(ii) unit vector
(iv) vector
2
2.
(i) Use a ruler and protractor to determine the magnitude (in centimeters) and
the direction of n .
Solution
Sketch the vector in standard
position and measure the
magnitude and direction. The
magnitude is 2 centimeters,
and the direction is 20.
(ii) Find the sum of v and w using:
a.
the parallelogram method.
b.
the triangle method.
c.
Compare the resultants found in both methods.
Solution
a. Copy v then copy w placing the initial
points together.
Form a parallelogram that has the two
vectors as two of its sides. Draw dashed
lines to represent the other two sides.
The resultant is the vector from the vertex
of v and w to the opposite vertex of the
parallelogram.
b. Copy v then copy w so that the initial
point of w is on the terminal point of v .
The resultant is the vector from the initial
point of v to the terminal point of w .
c. Use a ruler and protractor to measure the magnitude and direction of each resultant. The resultants of both
methods have magnitudes of 2.6 centimeters and directions of 42. So, the resultants found in both methods
are equal.
3
(iii) RECREATION A hang-glider traveled forward at 4 m/s and descended at 2 m/s. Determine the
magnitude of the resultant velocity of the hang-glider.
Solution
Let 1 centimeter represent 2 m/s. Draw two vectors, f
and d , to represent the forward velocity and the
descending velocity of the hang-glider, respectively.
Use the parallelogram method. Copy the forward
velocity vector. Then copy the descending velocity
vector placing the initial point at the initial point of the
forward velocity vector. Draw dashed lines to
represent the other two sides of the parallelogram.
The resultant velocity of the hang-glider is the vector
from the vertex of the two original vectors to the
opposite vertex of the parallelogram. Measure the
resultant, 2.3 centimeters. Multiply the resultant’s
magnitude by 2 to determine the magnitude of the
resultant velocity of the hang-glider.
2.3 cm  2 m/s = 4.6 m/s
The hang-glider is moving at 4.6 m/s.
(iv) Use the triangle method to find 2 v - 3 w .
Solution
Rewrite the expression as a sum.
2 v - 3 w = 2 v + (-3 w )
Draw a vector twice the magnitude of v to
represent 2 v . Draw a vector with the
opposite direction to w and three times its
magnitude to represent 3 w .
Place the initial point of the second vector
on the terminal point of the first vector.
(Tip-to-tail method)
The resultant vector is the answer.
4
Chapter 8.2 Algebraic Vectors
3.
(i) Write the ordered pair that represents the vector from X(-2, 4) to Y(4, -6). Then find the magnitude
of XY .
(ii) Let m = 2, -3, n = 1, 5, and p = -2, 4. Find each of the following.
a.
n +p
Answer ……………………….
c. 3 n
b. m - p
Answer……………………………….
d. 2 m + 3 p
Answer……………………………….
Answer……………………………….
(iii) Write AB as the sum of unit vectors for A(3, -2) and B(7, 4).
Answer……………………………….
Chapter 8.3 Vectors in Three-Dimensional Space
4.
(i)Write the ordered triple that represents the vector from X(4, 2, -5) to Y(3, -4, 1).
Answer……………………………….
(ii) Suppose the flight of a baseball passed through points at (2, 6, 9) and (8, 4, 7), in which each unit
represents a meter. What is the magnitude of the displacement the baseball experienced in traveling
between these two points? Round to the nearest tenth.
Answer……………………………….
(iii) Find an ordered triple that represents 4 p - 3 q if p = 2, 5, 3 and q = (4, -2, 1).
Answer……………………………….
5
(iv) Write AB as the sum of unit vectors for A(4, 2, 6) and B(-3, 8, -1).
Answer……………………………….
Chapter 8.4 Perpendicular Vectors
(i) Find each inner product if p = (3, 7), q = (2, -3), and m = (9, 6). Are any pairs of vectors
perpendicular?
a.
p q
Answer……………………
b. p  m
c.
q m
Answer……………………… Answer………………
(ii) Find the inner product of a and b if a = -4, 2, 5 and b = 3, 6, 1. Are the two vectors
perpendicular?
Answer……………… ………………
(iii) Find the cross product of v and w if v = 5, 2, 3 and w = -2, 5, 0. Verify that the resulting
vector is perpendicular to the two original vectors.
Answer……………………………
Chapter 8. 6 Vectors and Parametric Equations
5.
(i) Write a vector equation describing a line passing through P1(3, 2) and parallel to a = 4, -1.
6
Answer………………………………
(ii) Find the parametric equations for a line parallel to q = 4, -2 and passing through the point at (-1,
-3).
Answer………………………………
(iii) Write parametric equations of y = 3x – 5.
Answer………………………………
(iv) Write an equation in slope-intercept form of the line whose parametric equations are
x = 3 + 2t and y = -1 – 4t.
Answer………………………………
Good Luck
7