Introduction to Vectors
▪ Vector definition and representation
▪ Types of vectors
▪ Elementary vector operations – addition
▪ Elementary vector operations – subtraction
▪ Vector components
▪ 3D vectors and direction cosines
▪ Right-handed coordinate system
AMC511S – Engineering Mechanics 114 (Statics) | Andrew Zulu
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Introduction to Vectors
Scalar – quantity with only magnitude e.g. mass
Vector – quantity with magnitude and direction e.g. force
Tensor – of zeroth, first, second order (scalar, vector, matrix, resp.)
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Vector representation: vector 𝐴𝐵; 𝐴𝐵; AB (bold) | Unit/basis vectors 𝑖෠, 𝑗෡, 𝑘.
Vector types – free, sliding, fixed
> Free vector – action not confined to unique line in space e.g. moment vector.
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Vector types
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> Sliding vector – unique line in space but no unique point of application e.g. force on rigid
body
> Fixed vector – unique point of application e.g. force on non-rigid body.
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Elementary vector operations
> Addition – parallelogram, triangle, polygon rules
𝑢 + 𝑣റ = 𝑤
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Elementary vector operations
> Subtraction – parallelogram, triangle, polygon rules
𝑢 − 𝑣റ = 𝑤
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Null vector / vector components
> Null vector (equal and opposite) 𝑢 − 𝑣റ = 0
> Vector components (2D): orthogonal/rectangular;
acute & obtuse
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3D vector components / direction cosines
Vector components: Orthogonal/rectangular (3D)
𝐹റ = 𝐹𝑥 𝑖+Ƹ 𝐹𝑦 𝑗+Ƹ 𝐹𝑧 𝑘෠ | 𝐹 =
𝐹𝑥2 + 𝐹𝑦2 + 𝐹𝑧2 | Direction cosines: 𝑐𝑜𝑠𝜃𝑥 = 𝑙 = 𝐹𝑥 /𝐹 ;
𝑐𝑜𝑠𝜃𝑦 = 𝑚 = 𝐹𝑦 /𝐹; 𝑐𝑜𝑠𝜃𝑧 = 𝑛 = 𝐹𝑧 /𝐹; 𝑙2 + 𝑚2 + 𝑛2 = 1 .
Right-handed coordinate system: +ve (ccw)
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