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In this chapter on Vector Algebra, we delve into the definitions and properties of scalars and vectors, exploring null vectors, equal vectors, collinear vectors, unit vectors, and more. The laws of Vector Algebra are proven, and problems involving vector addition and subtraction are solved. Examples illustrate the concepts discussed.
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Welcome To My Presentation Hossain Md Waseem Firoz Instructor (Mathematics) 1stShift Kushtia Polytechnic Institute
Chapter 13 (Vector Algebra) Define Scalar and Vector. Explain Null vector, Equal vector, Collinear vector, Unit Vector, Linear Combination, Direction Cosines and Direction ratios, Dependent and Independent Vectors. Prove the Laws of Vector Algebra. Solve Problems involving Addition and Subtraction of Vectors.
Scalar Vector In mathematics and physics, a scalar is a quantity that only has magnitude (size), while a vector has both magnitude and direction. Examples of scalar quantities include pure numbers, mass, speed, temperature, energy, volume, and time. Examples of vector quantities include velocity, acceleration, momentum, displacement, and forces, such as weight and friction.
Scalar Vector Examples of Scalars Here are some examples of scalar quantities: Mass Speed Length Volume Density Power Pressure Temperature
Vector In mathematics and physics, a scalar is a quantity that only has magnitude (size), while a vector has both magnitude and direction. Examples of scalar quantities include pure numbers, mass, speed, temperature, energy, volume, and time. Examples of vector quantities include velocity, acceleration, momentum, displacement, and forces, such as weight and friction.
Vector Examples of Vectors Here are some examples of vector quantities: Force Weight Friction Acceleration Momentum
Definitions Null Vector: The null vector is defined to have zero magnitude and no particular direction. If two vectors are perpendicular to each, the magnitude of their cross product is equal to the product of their magnitudes. since the sine of 90 is equal to unity. (4.47)
Equal vector Equal vectors are vectors that have the same magnitude and the same direction. Equal vectors may start at different positions. Note that when the vectors are equal, the directed line segments are parallel.
Collinear Vector Vectors that lie along the same line or parallel lines are known to be collinear vectors. They are also known as parallel vectors. Two vectors are collinear if they are parallel to the same line irrespective of their magnitudes and direction.
Unit Vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in. (pronounced "v-hat").
Linear Combination In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).
Direction Cosines and Direction ratios Direction ratios are the components of a vector along the x- axis, y-axis, and z-axis, and direction cosine is the cosine of the angle subtended by this line with the x-axis, y-axis, and z- axis. May 4, 2023 The core concepts of three-dimensional geometry are direction cosines and direction ratios. What are direction cosines of a line that passes through the origin that makes angles with the coordinate axes? This lesson helps you understand the concepts of direction cosines and direction ratios which are nothing but numbers proportional to the direction cosines. A solved problem, in the end, will help you understand the concepts better.
Dependent and Independent Vectors A set of vectors is linearly dependent if there is a nontrivial linear combination of the vectors that equals 0. A set of vectors is linearly independent if the only linear combination of the vectors that equals 0 is the trivial linear combination (i.e., all coefficients = 0).
Prove the Laws of Vector Algebra What are the laws of vector algebra? v+w=w+v Commutative Law. u+(v+w)=(u+v)+w Associative Law. v+0=v=0+v Additive Identity. v+( v)=0 Additive Inverse. k(lv)=(kl)v Associative Law. k(v+w)=kv+kw Distributive Law. (k+l)v=kv+lv Distributive Law.
Solve Problems involving Addition and Subtraction of Vectors.
Solve Problems involving Addition and Subtraction of Vectors.
