MTH405_Wk_5_Lect_2 - MTH405 Wk 5 Lect 2 Vectors Scalar and...

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Unformatted text preview: MTH405 Wk 5 Lect 2 Vectors Scalar and Vector Quantities There are alot of scalar and vector quantities used in Physics and Engineering. We will discuss some of it here. Denition. 1. A scalar quantity is referred to a value which has a magnitude only. A vector quantity is referrred to a value which has both a magnitude and a direction. Example. Some examples of scalar and vector quantities. • Scalar - length, area, volume, speed, mass, density, pressure, temperature, energy, entropy, work, power, etc. 2. Vector - displacement, direction, velocity, acceleration, momentum, force, lift, drag, thrust, weight. Example. Determine which of these are scalar and which are vector quantities 1. The weight of rice is 50kg. • 2. The house is10m away. 3. The building is 10m away due east. 4. A car is travelling at60m/s. 5. A car is travelling at 60m/s travelling towards Lami. 6. 7. √ 12m 2ˆi + 2ˆj + 2kˆ m in 3D-space 1 Measure of Vector Quantities Notice that a vector has both magnitude and direction. How can be describe a vector? • Well, magnitude can be simply described using numbers (or scalar values, just like scalar quantity is described) • But for direction, we have to use spacial conveniently, known as dimensions . • Since we live in 3 dimensional space, we use the 3 vectors to describe the 3 directions that we have. These 3 vectors are known as unit vectors. quantities or more Unit Vectors Denition. A unit vector is dened as a vector which describes a direction but has a magnitude of 1. (This magnitude will later be known as norm which is the length of a vector) • This means when using unit vectors to describe normal vectors, the unit vectors contribute to the direction but do not contribute anything to the magnitude of the vector. • In the 3 dimensional Cartesian plane, the 3 unit vectors that we will use are: positive direction x-axis = ˆi positive direction y -axis = ˆj positive direction z -axis = kˆ Using this, we can now give a complete denition of a vector: Denition. Let a, b, c ∈ R. Then a is dened by vector ~v in 3-dimensional ˆ ~v = aˆi + bˆj + ck. In matrix terminology, the similar vector is given by a b . c The quantities a, b, c are the vector components 2 of vector ~v . space • A vector is a special type of matrix (it has only 1 column but many rows). • Note that ˆi, ˆj and kˆ are orthogonal to each other. This means their direction is 90◦ to each other. This otrthogonality will allow us to nd dot product and cross product of vectors later on. Vector Algebra Vector Algebra includes 3 operations which are possible on vectors. These 3 operations are: • Vector Negation • Vector Addition • Scalar Multiplication of Vectors The word 'Algebra' in Vector Algebra does not come out of nowhere: the possibility of conducting these operations (negation, addition, scalar multiplication) satises the denition of a Space in Mathematics. Hence, the set of vectors becomes a Vector Space just like R, the set of real numbers. Doing addition, negation and multiplication in R, we say we are doing algebra in R. Similarly, doing these operations in a vector space, we say that we are doing Vector Algebra. Remark. 3 Negation of a Vector Given a vector ~v1 ˆ a1ˆi + b1 ˆj + c1 k, = its negation is ˆ (−1)a1ˆi + (−1)b1 ˆj + (−1)c1 k,       = a1 −ˆi + b1 −ˆj + c1 −kˆ (−1) · ~v1 = −~v1 where −ˆi, −ˆj , −kˆ is the negative direction of their axis. Vector Addition Given two vectors ~v1 = ~v2 = a1ˆi + b1 ˆj + c1 kˆ ˆ a2ˆi + b2 ˆj + c2 k, we take its sum ~v1 + ~v2 ˆ (a1 + a2 )ˆi + (b1 + b2 )ˆj + (c1 + c2 )k. = Scalar Multiplication of a Vector Given a scalar d∈R and a vector ~v1 = ˆ a1ˆi + b1 ˆj + c1 k, its scalar multiplication d · ~v1   = d · a1ˆi + b1 ˆj + c1 kˆ = ˆ (da) ˆi + (db) ˆj + (dc) k. 4...
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