Lecture 3: Vectorsweb.mst.edu/~vojtaa/engphys1/lectures/lec03.pdf · •Unit vector notation •...
Transcript of Lecture 3: Vectorsweb.mst.edu/~vojtaa/engphys1/lectures/lec03.pdf · •Unit vector notation •...
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• Definition
• Graphical addition and subtraction of vectors
• Unit vector notation
• Vector components, magnitude and direction
• Addition and subtraction of vectors in unit vector notation
Lecture 3: Vectors
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Vectors
A vector is a quantity that has size (magnitude)
and direction. It can be symbolized by an arrow.
*Sometimes bold-face type also indicates a vector – hard to do in
handwriting
Notation convention:Ԧ𝐴 denotes vector of magnitude 𝐴 = | Ԧ𝐴|
Length of the arrow
represents magnitude
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Vector addition - graphically
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Vector subtraction - graphically
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Unit vectors
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Unit vector notation
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Vector components
𝐴𝑥 = +𝐴𝑐𝑜𝑠θ𝐴𝑦 = +𝐴𝑠𝑖𝑛θ
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Vector components
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Unit vector notation
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Magnitude and direction
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Example
A displacement of 5 km is directed θ = 53° East of
South. What is the displacement vector in unit-vector
notation?
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Example
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Example
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Vector addition in components
Ԧ𝐶 = 𝐴𝑥 Ƹ𝑖 + 𝐴𝑦 Ƹ𝑗 + (𝐵𝑥 Ƹ𝑖 + 𝐵𝑦 Ƹ𝑗)
Ԧ𝐴 = 𝐴𝑥 Ƹ𝑖 + 𝐴𝑦 Ƹ𝑗
𝐵 = 𝐵𝑥 Ƹ𝑖 + 𝐵𝑦 Ƹ𝑗
Ԧ𝐶 = (𝐴𝑥+𝐵𝑥) Ƹ𝑖 + (𝐴𝑦 + 𝐵𝑦) Ƹ𝑗
𝐶𝑥 = 𝐴𝑥 + 𝐵𝑥𝐶𝑦 = 𝐴𝑦 + 𝐵𝑦
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Vector subtraction in components
𝐷 = 𝐴𝑥 Ƹ𝑖 + 𝐴𝑦 Ƹ𝑗 − (𝐵𝑥 Ƹ𝑖 + 𝐵𝑦 Ƹ𝑗)
Ԧ𝐴 = 𝐴𝑥 Ƹ𝑖 + 𝐴𝑦 Ƹ𝑗
𝐵 = 𝐵𝑥 Ƹ𝑖 + 𝐵𝑦 Ƹ𝑗𝐷 = Ԧ𝐴 − 𝐵
𝐷 = (𝐴𝑥−𝐵𝑥) Ƹ𝑖 + (𝐴𝑦 − 𝐵𝑦) Ƹ𝑗
𝐷𝑥 = 𝐴𝑥 − 𝐵𝑥𝐷𝑦 = 𝐴𝑦 − 𝐵𝑦
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Example
Express the vectors Ԧ𝐶 = Ԧ𝐴 + 𝐵
and 𝐷 = 𝐵 − Ԧ𝐴 in unit vector
notation in terms of 𝐴, 𝐵, and .
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Example with tilted coordinate system
x
y
W θ
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x
y
W θ
θ
θ
𝑁𝑥 = 0𝑁𝑦 = 𝑁
𝑊𝑥 = 𝑊 sin θ𝑊𝑦 = −𝑊 cos θ
𝑃𝑥 = −𝑃 𝑐𝑜𝑠 θ𝑃𝑦 = −𝑃 𝑠𝑖𝑛 θ