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Vectors Objectives: • Define the concept of a vector • Learn how to perform basic vector operations using graphical and numerical methods • Learn how to use vector algebra to solve simple problems VectorA vector is a quantity that has:• a magnitude• a direction(e.g. change in position) VA scalar quantity has magnitude only (e.g. time) 1
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Numerical RepresentationMethods of expressing a vector (V) numerically:• its magnitude (V) and direction (θ) with respect to a reference axis• its components (Vx, Vy) along each reference axis y Vy V V θ x (0,0) Vx Vector Composition• Process of determining a single vector from two or more vectors by vector addition• Performed graphically using tip-to-tail method V1 + V2 copy of V2 V2 V1resultant : vector resulting from the composition 2
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Vector Resolution• Process of replacing a single vector with two perpendicular vectors whose composition equals the original vector V2 V V2 V V1 V1 Another Resolution of V Resolution into Components• Trigonometry can be used to numerically resolve a vector into its x- and y-components y Vx cos θ = V V Vy sin θ = V V Vy Vx = V * cos θ θ Vy = V * sin θ x(0,0) Vx 3
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Composition of Components• A vector can be numerically composed from its components using geometry and trigonometry y V V = V x2 + Vy2 Vy V Vy θ = atan Vx θ x (0,0) Vx Composition of 1-Dimensional Vectors• Vectors pointing in same direction: V1 V2 • magnitudes sum, V1 + V2 • direction remains same• Vectors pointing in opposite direction: V1 V2 V1 + V2 • magnitudes subtract: (larger – smaller), V1 • direction is that of larger vector V1 + V2 V2 4
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Numerical Vector Composition 1. Draw x- and y-axes 2. Resolve each vector into x and y components 3. x component of resultant = add each component pointing in +x direction and subtract each component pointing in –x direction. 4. y component of resultant = add each component pointing in +y direction and subtract each component pointing in –y direction. 5. Draw the x and y components of the resultant 6. Compose the resultant from its components Example VR y V2 V2 VRyV2y = V 2 sin θ2 V1 θR V1y = V 1 sin θ1 θ2 θ1 x V2x = V 2 cos θ2 VRx V1x = V 1 cos θ1 VRx = V1x – V2x VR = VRx2 + VRy2 VRy = V1y + V2y θR = atan (VRy / VRx ) 5
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Alternate Method of Composition1. Draw vectors “tip-to-tail”2. Draw resultant vector to form a triangle3. Draw x- and y-axes at tail of first vector4. Determine the angle between the first and second vector in the triangle.5. Use Law of Cosines to determine the magnitude of the resultant.6. Use Law of Sines to determine the angle between the first vector and the resultant7. Compute direction of the resultant from identified angles Exampley V2 V2 VR α sin α sin β β = VR V2 θR V1 V2 sin α θ1 β = asin x VR VR = V 12 + V22 – 2V1V2 cos α θR = θ1 + β 6
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Vector-Scalar MultiplicationIf a vector V is multiplied by a scalar n:• If n > 0: – magnitude of resultant = n * V – direction of resultant = direction of V• If n < 0: – magnitude of resultant = (–n) * V – direction of resultant = opposite direction of V 3*V V θ θ -1 * V Vector Subtraction • Subtraction of a vector performed by adding (–1) times the vector • Can be performed graphically or numerically V2 V1 -1 * V2 -1 * V2 V1 – V2 7
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Subtraction as a Change• Subtraction can be pictured as the difference or change between two vectors that originate from the same point y V1 + (V2 – V1) = V2 V1 V2 – V1 V2 x Graphical Solution Using Vectors1. Establish a scaling factor for the graph (e.g. 1cm = 10 m/s)2. Carefully draw vectors with the correct length (based on the scaling factor) and direction3. Use graphical methods of composition, resolution, scalar multiplication, and/or subtraction to find desired resultant4. Carefully measure the length and direction of the resultant.5. Use scaling factor to convert measured length to magnitude 8
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Example Problem #1Two volleyball players simultaneously contact the ball above the net.Player #1 hits the ball from the left with a force of 300 N (67 lb), angled 45° below the horizontal.Player #2 hits the ball from the right with a force of 250 N (56 lb), angled 20° below the horizontal.What is the magnitude and direction of the net force applied to the ball by the 2 players? Numerical Solutions Using Vectors1. Sketch the vectors on a diagram of the problem2. Choose and diagram the coordinate axes, based on: • axes used in the problem statement • axes that are physically meaningful3. Establish and label known magnitudes and angles or x- and y-components4. Use numerical methods of composition, resolution, scalar multiplication, and/or subtraction to find desired solution 9
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Graphical vs. Numerical Method• Graphical Method – Simple – Must be done by hand – Gives approximate result• Numerical Method – Requires complex calculations – Gives accurate result – Can be performed by computer – Can perform analyses in 3 dimensions Example Problem #2A golfer is teeing off from the center of the fairway for a hole that is located 300 yards away and 30° to the right of center.The golfer’s tee shot goes 210 yards and 15° to the left of center of the fairway.To reach the hole on the second shot, how far and in what direction must the golfer hit the ball? 10
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