Maths, Undergraduate, The Scalar Product, QMUL
- Read the task in Box 1.
- Download each of the pdfs and read them. Keep these open so you can refer to them as you watch the video.
- Watch the video.
The Mathematical Writing course at QMUL (http://www.maths.qmul.ac.uk/~fv/teaching/mw/ ) provides specific training on developing and presenting mathematical arguments. The course aims to facilitate students’ understanding of fundamental mathematical ideas through written explanation of mathematical concepts. This is particularly useful because at ‘A’ level students often use ‘templates’ or patterns to solve mathematical problems. While these templates may work in allowing students to solve a problem, they do not help with understanding. In the Mathematical Writing course, students engage with the underlying mathematical theory and learn the language of mathematics. Writing is used to help students consolidate, and perform a synthesis of, the material introduced in first year mathematics courses. It is also essential preparation for any student who wants to do a final year project.
In this task, the student is given two pages of mathematical description taken from a first year textbook. The topic is the scalar product, a fundamental concept in elementary geometry. The task is to provide a title for the extract; a short list of key points, and a summary. This is not easy; the student has to extract a few important points out of a large mass of details and examples and organise this information into an effective summary.
The task has to be completed without using any mathematical symbols. Having to express mathematics without symbols is a most useful exercise firstly because it brings about the discipline needed to use symbols effectively, and secondly because it is invaluable for learning how to communicate to an audience of non-experts.
Scalar product of two vectors and its properties.
In three-dimensional space, scalar product (sometimes called dot product) of two non-zero vectors is the product of their norms and cosine of the angle between them, which gives scalar result.
Following are some properties of scalar product:
- scalar product of a vector with itself is the square of its norm.
- If the angle between is acute, the scalar product is positive.
- If the angle is obtuse, the scalar product is negative.
- If the angle is right angle, the scalar product is zero and we can say two vectors are orthogonal (perpendicular)
- If the scalar product is zero, two vectors are not orthogonal as one of them could be zero vector.
- Scalar product operation is commutative as well as distributive over addition.
There is another way of compute scalar product of two vectors when they are expressed in terms of Cartesian coordinates. To do so we take product of each coordinate of the two vectors respectively, then adding them together which gives you the scalar product.
Scalar product of three-dimensional vectors.
1. Two ways of computing scalar products.
2. Finding an angle between two vectors.
Scalar (or dot) product of two non zero vectors in three-dimensional space is the product of their norms and the cosine of the angle between the directions of the vectors.
If one of the vectors is zero or the angle is a right angle (in which case the vectors are orthogonal), the scalar product is defined to be 0.
Dot product operation is commutative and distributive over addition. It is useful when we express vectors in terms of the standard unit vectors. After some rearrangements, we find other way of obtaining the dot product. Namely, it is a sum of products of vectors’ corresponding components.
Because we have two definitions of scalar product, we can use the two expressions to find the cosine of an angle between the vectors, and hence the angle itself.
The scalar product of vectors
Summary: The scalar product (or dot product) is an operation that takes any two vectors (the vectors can be equal) and produces a scalar. The operation can be deﬁned algebraically, where it is the sum of the products of the corresponding vector components, or, if the vectors are non-zero, geometrically, where it is expressed as the product of three quantities, namely the norm of each vector and the cosine of the angle between the vectors. By combining both deﬁnitions, one can compute this angle if one wishes. The scalar product obeys certain algebraic laws; in particular it enjoys commutativity and distributivity over addition, and these laws can be veriﬁed using the deﬁnition. A useful fact is that the scalar product of two vectors is zero, if and only if the vectors are orthogonal (i.e. perpendicular) or at least one of them is the zero vector.
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