MA10236 Methods and Applications 1B, 2020. Chris Budd
Worksheet 1: An introduction to vectors
This sheet is meant to give you a bit of a warm up and then some practice in vectors. The W questions are a warm up for discussion in your Tutorial in Week 1. You should then do all of the unstarred questions for your Tutorial in Week 2. The sheet also has starred (*) questions which are a little harder and are meant for revision. It also contains your first very optional * Challenge Question which is meant to give you something to think about. For a bit of light relief each sheet will have a highly optional Puzzle Corner. Have fun!
You should hand in your answers to your tutors on Tuesday 11th February.
Which of the following are vector or scalar quantities or other?
Mass, weight, acceleration, spin, surface area, the weather, personality, political views?
Let \(\mathbf{a}, \mathbf{b}, \mathbf{c}\) be three vectors. Draw a figure to show that \[(\mathbf{a} + \mathbf{b}) + \mathbf{c} = \mathbf{a} + (\mathbf{b} + \mathbf{c} ).\]
Let \(E\) and \(F\) be the midpoints of the sides \(AB\) and \(CD\) of the quadrilateral \(ABCD\). Prove that \[\overrightarrow{EF} = \frac 12 (\overrightarrow{BC} + \overrightarrow{AD}).\]
Use the fact that the length of one side of a triangle is less than or equal to the sum of the lengths of the other two sides to show that, for any vectors \(\mathbf{a}\) and \(\mathbf{b}\), \[| \, \|\mathbf{a}\| - \|\mathbf{b}\| \, | \le \|\mathbf{a} + \mathbf{b} \| \le \|\mathbf{a}|| + \|\mathbf{b}\|.\]
Assume that the vectors \(\mathbf{a}\) and \(\mathbf{b}\) are nonzero and are not parallel. If \(\mathbf{c}\) is in the plane of \(\mathbf{a}\) and \(\mathbf{b}\) draw a figure to show that it is possible to write \[\mathbf{c} = \lambda \mathbf{a} + \mu \mathbf{b}\] for some scalars \(\lambda\) and \(\mu\), and prove that such \(\lambda\) and \(\mu\) are found uniquely.
\(A, \, B, \, C, \, D\) are the vertices of a quadrilateral. Using position vectors, prove that if \(P\), \(Q\), \(R\) and \(S\) are the mid-points of the sides \(AB\), \(BC\), \(CD\) and \(DA\) respectively then \(PQRS\) is a parallelogram.
If \(\mathbf{a}\) and \(\mathbf{b}\) denote the position vectors of points \(A\) and \(B\) (with respect to the origin at \(O\)), show that the position vector \(\mathbf{p}\) of the point \(P\) which divides the line segment \(AB\) in the ratio \[AP : PB = \lambda : 1-\lambda,\] where \(0\,<\,\lambda\,<\,1\), is \[\mathbf{p} = (1-\lambda) \mathbf{a} + \lambda \mathbf{b}.\]
If \(\mathbf{a} = 2 \mathbf{i}-3\mathbf{j} + \mathbf{k}\) and \(\mathbf{b} = 3 \mathbf{j} + 4 \mathbf{k}\) find the value of \(\|(\mathbf{a}.\mathbf{b}) \; \mathbf{b}\|\).
Prove, using the dot product, that the angle between two diagonals of a cube is \(\cos ^{-1}(1/3)\).
If \(\mathbf{a} = 2 \mathbf{i} + 3 \mathbf{j} - 4 \mathbf{k}\) and \(\mathbf{b} = 2 \mathbf{i} - 3 \mathbf{j} + 5 \mathbf{k}\) find the value of the scalar \(\lambda\) such that \(\mathbf{a} - \lambda \mathbf{b}\) is orthogonal to \(\mathbf{a}\).
For what value of \(p\) does the vector \(p\mathbf{i} + 2 \mathbf{j}\) make an angle \(\pi/4\) with the vector \(3 \mathbf{i} + \mathbf{k}\)?
Let \(\mathbf{a}\) be a non-zero vector, and \(p\) be a scalar. Describe geometrically all the vectors \(\mathbf{x}\) such that \[\mathbf{x}.\mathbf{a} = p.\]
HINT. First take \(\mathbf{a} = {\mathbf i}\), then generalise.
Using the dot product, show that \[\|\mathbf{u} + \mathbf{v}\|^2 + \|\mathbf{u} - \mathbf{v}\|^2 = 2 (\|\mathbf{u}\|^2 + \|\mathbf{v}\|^2).\]
To find out more about the course, \(n\) students visit my office. I give each of them one different piece of information about the course. They then email each other. In each email the two students share all of the information that they know. How many emails are needed before each of the \(n\) students knows everything that the other students know?
This lecture course started on 3rd February. Why is this an important date in the history of rock and roll?
IMPORTANT NOTE AND A HORRIBLE WARNING: In all your work, you MUST show which variables are vectors and which are scalars or lengths. Writing \(AB\) means the line joining point \(A\) and point \(B\), or the distance between \(A\) and \(B\), whereas with an overhead arrow, \(\overrightarrow{AB}\) means the vector which starts at \(A\) and ends at \(B\). Writing \(a\) means a scalar variable; if you mean a vector, you MUST write a ‘twiddle’ sign underneath: \({\underset{\sim}{a}}\). In print, we use bold type to indicate a vector: \(\mathbf{a}\). This convention is used in the printed notes and the tutorial sheets. It also applies to \(0\): writing \(0\) means the (scalar) number zero, while \({\underset{\sim}{0}}\) or \(\mathbf{0}\) means the zero vector. In the lectures I will use ‘twiddles’ \({\underset{\sim}{a}}\), and in the sheets and solutions I will use bold-face,\(\quad \mathbf{a}\), \(\mathbf{0}\).
A common mistake made by (otherwise splendid) MA10236 students, is to try dividing a vector by a vector, i.e. treating a vector as if it were a scalar. (In later years you will find that this is OK, but you need to learn about the tensor product first.) You can avoid doing this if you carefully indicate which quantities are vectors. In the exam you will LOSE MARKS if you don’t indicate vectors correctly (even if your solution is otherwise correct).
CJB