School of Engineering, University of Warwick

ES157 - MATHEMATICS FOR ENGINEERS Study Notes for WEEK 4 VECTOR ALGEBRA I 1. Motivation Real numbers tell us only the size of some quantity. In engineering many properties have both a size and some form of direction or orientation. For example a force has a magnitude and direction, or a point in three-dimensional space is described by three coordinates. Such quantities are known as vectors or vector quantities. This week we start exploring how vectors can be used. Many applications such a computer graphics and the study of dynamics rely heavily on vectors. 2. Key Concepts The key concepts discussed in the readings are: • definition of a vector • vector addition and subtraction • scaling vectors • Cartesian component form of vectors • equation of a straight line in vector form • the definition of the scalar (dot) product of vectors • the equation of a plane in vector form 3. The skills to be developed At the end of this week’s work you should be able to: • manipulate vectors from simple vector diagrams • find the modulus of a vector • form a unit vector in the same direction as a vector • add and subtract vectors • carry out relative velocity calculations • express the equation of a straight line in standard form • from the standard form of the equation of a straight line find a point on the line and a vector parallel to the line (and vice-versa) • calculate the scalar product of two vectors • find the angle between two vectors • calculate the work done when a force moves its point of application • find the projection of one vector in the direction of another vector • calculate the equation of a plane from the normal vector and a point in the plane (and vice-versa) • find the angle between two planes • find the perpendicular distance of a point to a plane

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4. This week’s readings and practice exercises For this week’s work we move to Chapter 9 of the book. We will do the whole of the chapter; one aspect of vectors will be left until the Week 5 work. As in previous weeks, I am suggesting certain questions from the block exercises which serve to help you learn. You should do as many of these as you feel necessary to be confident that you have understood the topic. Again, it is the revision questions in §5 which will determine whether your understanding and skills are sufficient to be able to cope with examination-type questions. Chapter 9 Pages 494-506:

BLOCK 1: Basic concepts of vectors (Read the first part, pp. 494-496, quickly; vector addition, scaling and subtraction are more important; Also, resolving vectors is important, eg. for Statics)

Exercises:

Page 496, Q1; Page 499, Q1, Q2 Q3 (all easy!); Page 505, Q1, Q2, Q4; Page 506, Q1, Q2, Q3; Page 507 (End-of-block), Q4

Pages 508-523:

BLOCK 2: Cartesian components of vectors Note: this does NOT include §2.6. (The unit vectors, i and j, in the x and y directions; vectors in i-, j-component form; direction cosines (for interest only); three-dimensional (3D) coordinate frames; direction ratios and 3D direction cosines for interest only.)

Exercises:

Page 516, Q3, Q4, Q6; Page 521, Q1, Q2 (a), (b) & (c) only, Q3, Q4; 525 (End-of-block), Q1, Q3, Q4

Pages 526-535:

BLOCK 3: The scalar, or dot, product Note: Ignore example 3.15 (Definition, properties, dot-product formula for vectors in component form (note that the formula is in the Data Book, Page 10); angle between vectors; projection of a vector onto a different direction.)

Exercises:

Page 528, Q2, Q3, Q4, Q6; Page 530, Q1; Page 532, Q1, Q3, Q4 (important!); Page 533, Q1; Page 536, Q1, Q3; Page 537 (End-of-block), Q4, Q5, Q10

Page

MISS OUT BLOCK 4.... for the moment - we will return to it in Week 5 Pages 549-555:

BLOCK 5: Vector equation of a line and plane (Equation of a line in 3D space - vector and cartesian forms; equation of a plane - vector and cartesian forms

Exercises:

Page 553, Q1, Q2, Q3, Q4; Page 555, Q1, Q2 (also find the cartesian form of the plane)... Note: do not attempt Q3; Page 556 (End-of-block), Q9.

Supplement to Block 5 We can generate the cartesian form of a plane as follows. Using Fig. 5.2 on Page 554, we write the key vectors in component form: r = xi + yj + zk

where (x, y, z) is any point on the plane

a = a1 i + a2 j + a3 k

for example: a = 2i − j + 4k

S1b

n = n1 i + n2 j + n3 k

for example: n = 4i + 3j + k

(S1c)

2

(S1a)

The vector equation of a plane (top of Page 555) is r·n=a·n

(S2)

An so, substituting Eqns. S1 into each vector of Eqn. S2 and using the dot-product formula, we get n 1 x + n 2 y + n 3 z = a1 n 1 + a2 n 2 + a3 n 3 (S3a) which, using the example vectors in Eqns. S1, gives: 4x + 3y + z = 8 − 3 + 4 = 9

(S3b)

Equation(s) S3 show the standard cartesian form of the equation of a plane. Note that the coefficients of x, y and z in the standard form give the i, j and k coefficients of the normal to the plane. For example, given the plane 6x − y + 3z = −8 we could immediately deduce that the normal, n, to this plane is the vector 6i − j + 3k.... or any scalar multiple of this vector (including the negative - scalar multiple −1). You might also realise that ‘removing’ the z-dependence, say from Eqn. S3b, yields the equation of a line, i.e. 4x + 3y = 9, which upon re-arranging is y = − 43 x + 3. Obviously a plane collapses down to a line in two-dimensions; in fact this line is the intersection of the plane given by Eqn. S3b and the plane z = 0. Similarly, the intersection with, for example, the plane z = 1 would yield a line, this time given by 4x + 3y + 1 = 9 (or, upon re-arrangement, y = − 34 x + 83 ). 5. Revision questions for tutorials At the completion of the above learning, now attempt each of the following revision questions which relate to all the aspects of vectors covered this week. Your tutor will expect you to be familiar with (either have done or attempted) these questions because they will be raised at your Week 5 tutorial. Q1 (a) Find the vector (in the broken line) joining point O to N in Fig. E1a. N is the mid-point of AB. (b) The triangle of Fig. E1a is now extended into three dimensions to give the tetrahedron of Fig. E1b. Note that M is the mid-point of CB and the point P is the mid-point of M N . Find the vector joining O to P . [Ans. (a) 21 (a + b),

Fig. E1a

(b) 14 (a + 2b + c)]

Fig. E1b

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Q2 Two points, A and B, in three-dimensional space are respectively given by the cartesian coordinates (3, 0, −1) and (1, −2, 4). (a) Write down the position vectors (call these a and b) of the two points. [Ans. a = 3i − k, b = i − 2j + 4k] ~ (‘from A to B’). (b) Find the vector AB [Ans. −2i − 2j + 5k] (c) Find the unit vectors in the directions of a and b. [Ans. ˆ a=

√1 (3i 10

ˆ= − k), b

√1 (i 21

− 2j + 4k)]

[Ans.

√ 33 units]

(d) Find the distance from A to B.

(e) Find the cartesian equation of the straight line through A and B. y z+1 [Ans. x−3 −2 = −2 = 5 (or with any constant added to all three parts of the equation)] (f) Find a point on the line of part (e) which is 10 units away from A along the line in the direction of B. [Ans. (3 − √2033 )i − ( √2033 )j + (−1 + √5033 )k)] Q3 A ship imaginatively named A is travelling at 16 knots in a direction 60◦ East of due North. Concurrently a ship surprisingly named B steams at 20 knots in a direction 60◦ South of due East. Draw a velocity vector diagram for the two ships and find the velocity of B relative to A. [Ans. 25.6 knots in direction 81.3◦ South of due West] Q4 Two vectors are given by: a = 2i − 3j + k

b = −3i + 2j + 6k

(a) Find a · b (b) Find the angle between a and b [Ans. (a) −6, (b) 1.80 rads (103.2◦ )] Q5 A force, F = 5i − j + 5k Newtons, moves an object 10 metres in the direction s = 3i + k. Find the magnitude of the projection of F in the direction of s. Find the work done by the force.

√ √ [Ans. (a) 2 10 N, (b) 20 10 J]

Q6 A plane has a normal vector, n = 7i − j + 3k, and a point on the plane is (1, 2, −1). Find the cartesian equation of the plane. [Ans. 7x − y + 3z = 2] Q7 Two planes are given by the equations: 3x + y − 4z = 4

(plane1)

−x + y + 2z = 2

(plane2)

Find the line of intersection of these planes. y−(5/2) z−0 [Ans. x−(1/2) (3/2) = −(1/2) = 1 (or with any constant added to all three parts of the equation)]

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Q8 Find the angle between the planes 1 and 2 given in question 7 [Ans. 36.8◦ (or 180◦ − 36.8◦ = 143.2◦ )] Q9 (More difficult) A plane is given by x + 2y − 2z = 9 Find the (shortest) distance of the point B at (2, 3, −5) from this plane. ~ and using this form (Hint: Find a point, say A, on the plane. Then consider the vector AB a right-angled triangle with the perpendicular from the plane to the point B.) [Ans. 3 units]

ADL September 2000, revised PJT September 2001

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