2 Common Ways to Express Vectors Using Magnitude and Direction example d = 5m[ E37°N ] Using Components example d = (4,3) These two examples express the.

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Presentation transcript:

2 Common Ways to Express Vectors Using Magnitude and Direction example d = 5m[ E37°N ] Using Components example d = (4,3) These two examples express the SAME vector using two different methods! d = 5m[ E37°N ] = (4,3)

A sketch makes this easier to visualize d = 5m[ E37°N ] = (4,3)

A sketch makes this easier to visualize Draw and scale the axes d = 5m[ E37°N ] = (4,3)

A sketch makes this easier to visualize Draw and scale the axes Add the direction indicators North d = 5m[ E37°N ] = (4,3) East

A sketch makes this easier to visualize Draw and scale the axes Add the direction indicators Draw the vector from the origin North d = 5m[ E37°N ] = (4,3) East 37º d

A sketch makes this easier to visualize Draw and scale the axes Add the direction indicators Draw the vector from the origin show the components on the axes North d = 5m[ E37°N ] = (4,3) East 37º d (4,3)

First: From Magnitude and Direction notation into Component notation Converting Between Vector Notations

Magnitude & Direction to Components Draw the axes Example V=6.0m/s[E33°N]

Magnitude & Direction to Components Draw the axes Add the direction indicators North(y) East(x) Example V=6.0m/s[E33°N]

Magnitude & Direction to Components Draw the axes Add the direction indicators Draw the vector from the origin North East 33º V Example V=6.0m/s[E33°N]

Magnitude & Direction to Components Draw lines from the tip of the vector to each axis as shown (perpendicular) North East 33º V Example V=6.0m/s[E33°N]

Magnitude & Direction to Components Draw lines from the tip of the vector to each axis as shown (perpendicular) Use trigonometry (sine and cosine functions) to find the components as follows... North East 33º V Example V=6.0m/s[E33°N]

Magnitude & Direction to Components Use the triangle with the angle you already have North East 33º Example V=6.0m/s[E33°N] V

Magnitude & Direction to Components Use the triangle with the angle you already have Label the “x” and “y” sides North East 33º Example V=6.0m/s[E33°N] y x V

Magnitude & Direction to Components Use the triangle with the angle you already have Label the “x” and “y” sides Write in the value of the hypotenuse North East 33º 6.0 Example V=6.0m/s[E33°N] y x

Magnitude & Direction to Components Use the triangle with the angle you already have Label the “x” and “y” sides Write in the value of the hypotenuse Use the trig ratios sin = opp/hyp cos = adj/hyp North East 33º 6.0 Example V=6.0m/s[E33°N] y x

Magnitude & Direction to Components sin = opp/hyp North East 33º 6.0 Example V=6.0m/s[E33°N] y x

Magnitude & Direction to Components sin = opp/hyp sin33 = y / 6.0 North East 33º 6.0 Example V=6.0m/s[E33°N] y x

Magnitude & Direction to Components sin = opp/hyp sin33 = y / 6.0 y = (6.0)sin33 North East 33º 6.0 Example V=6.0m/s[E33°N] y x

Magnitude & Direction to Components sin = opp/hyp sin33 = y / 6.0 y = (6.0)sin33 y = (6.0)(0.5446) North East 33º 6.0 Example V=6.0m/s[E33°N] y x

Magnitude & Direction to Components sin = opp/hyp sin33 = y / 6.0 y = (6.0)sin33 y = (6.0)(0.5446) y = 3.3 North East 33º 6.0 Example V=6.0m/s[E33°N] y x

Magnitude & Direction to Components cos = adj/hyp North East 33º 6.0 Example V=6.0m/s[E33°N] y x

Magnitude & Direction to Components cos = adj/hyp cos33 = x / 6.0 North East 33º 6.0 Example V=6.0m/s[E33°N] y x

Magnitude & Direction to Components cos = adj/hyp cos33 = x / 6.0 x = (6.0)cos33 North East 33º 6.0 Example V=6.0m/s[E33°N] y x

Magnitude & Direction to Components cos = adj/hyp cos33 = x / 6.0 x = (6.0)cos33 x = (6.0)(0.8386) North East 33º 6.0 Example V=6.0m/s[E33°N] y x

Magnitude & Direction to Components cos = adj/hyp cos33 = x / 6.0 x = (6.0)cos33 x = (6.0)(0.8386) x = 5.0 North East 33º 6.0 Example V=6.0m/s[E33°N] y x

Magnitude & Direction to Components So the tip of the vector can be represented by the ordered pair (5.0,3.3) North East 33º 6.0 Example V=6.0m/s[E33°N] V= (5.0,3.3)

Magnitude & Direction to Components So the tip of the vector can be represented by the ordered pair (5.0,3.3) This is the same vector expressed using two different notations! North East 33º 6.0 Example V=6.0m/s[E33°N] V= (5.0,3.3)

Second: From Component notation into Magnitude and Direction notation Converting Between Vector Notations

Components to Magnitude & Direction Draw and scale the axes Example V=(5.0,3.3)

Components to Magnitude & Direction Draw and scale the axes Find the tip of the vector by plotting the ordered pair that represents the components Example V=(5.0,3.3) ( )

Components to Magnitude & Direction Draw the vector from the origin to the point plotted V Example V=(5.0,3.3) ( )

Components to Magnitude & Direction Draw the vector from the origin to the point plotted Use only one of the two triangles shown V Example V=(5.0,3.3) ( )

Components to Magnitude & Direction Draw the vector from the origin to the point plotted Use only one of the two triangles shown V Example V=(5.0,3.3) ( )

Components to Magnitude & Direction Draw the vector from the origin to the point plotted Use only one of the two triangles shown Identify lengths of the sides of the triangle V Example V=(5.0,3.3)

Components to Magnitude & Direction Use the pythagorean theorem to find the length of the vector required V Example V=(5.0,3.3)

Components to Magnitude & Direction Use the pythagorean theorem to find the length of the vector required V 2 = V Example V=(5.0,3.3)

Components to Magnitude & Direction Use the pythagorean theorem to find the length of the vector required V 2 = V 2 = V Example V=(5.0,3.3)

Components to Magnitude & Direction Use the pythagorean theorem to find the length of the vector required V 2 = V 2 = V 2 = 35.9 V Example V=(5.0,3.3)

Components to Magnitude & Direction Use the pythagorean theorem to find the length of the vector required V 2 = V 2 = V 2 = 35.9 V = Example V=(5.0,3.3)

Components to Magnitude & Direction Use the tangent function to find the angle at the base of the vector V Example V=(5.0,3.3)

Components to Magnitude & Direction Use the tangent function to find the angle at the base of the vector tan  = opp / adj V Example V=(5.0,3.3)

Components to Magnitude & Direction Use the tangent function to find the angle at the base of the vector tan  = opp / adj tan  = 3.3 / 5.0 V Example V=(5.0,3.3)

Components to Magnitude & Direction Use the tangent function to find the angle at the base of the vector tan  = opp / adj tan  = 3.3 / 5.0 tan  = 0.66 V Example V=(5.0,3.3)

Components to Magnitude & Direction Use the tangent function to find the angle at the base of the vector tan  = opp / adj tan  = 3.3 / 5.0 tan  = 0.66  = tan -1 (0.66) V Example V=(5.0,3.3)

Components to Magnitude & Direction Use the tangent function to find the angle at the base of the vector tan  = opp / adj tan  = 3.3 / 5.0 tan  = 0.66  = tan -1 (0.66)  = 33° V Example V=(5.0,3.3) º

Components to Magnitude & Direction North East 33º V Example V=6.0m/s[E33°N] V= (5.0,3.3) ( ) Now add the direction indicators and express the angle correctly

Components to Magnitude & Direction North East 33º V Example V=6.0m/s[E33°N] V= (5.0,3.3) ( ) Now add the direction indicators and express the angle correctly  = [E33°N]

Usefulness of Component Notation Vector operations such as addition, subtraction and scalar multiplication become simple using component notation!

Vector Addition Using Components Simply add the “x” components to get the “x” component of the answer Add the “y” components to get the “y” component of the answer example: (2,3) + (6,3) = (2+6, 3+3) = (8,6) adding multiple vectors is also simple: (4,-1) + (-5,2) + (6,3) + (5,2) = (10,6)

Subtraction Using Components Simply subtract the “x” components to get the “x” component of the answer Subtract the “y” components to get the “y” component of the answer example: (2,3) - (6,3) = (2-6, 3-3) = (-4,0) subtracting multiple vectors is similar: (4,-1) - (-5,2) - (6,3) = (3,-6)

Scalar Multiplication Using Components Simply multiply the “x” and “y” components separately to get the “x” and “y” components of the answer example: if d = (2,3) then 4d = (8,12)

Solving Problems Using Components Since the vector operations are much simpler using components, it is often useful to convert vectors into component notation then solve the vector operations and then convert back to traditional magnitude and direction notation