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Vectors and Vector Addition Honors/MYIB Physics

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This is a vector.

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It has an x-component and a y-component.

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The x-component is 3 units, and the y-component is 2 units. 3 units 2 units

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Here is a second (red) vector.

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Its x-component is 1 unit and its y-component is 4 units. 1 unit 4 units

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I can add the two vectors by drawing them head-to-tail.

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The sum is called the resultant vector. It is drawn from start to end.

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The resultant has an x- component of 4 units and a y- component of 6 units. 4 units 6 units

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This can be found by adding the two x-components and adding the two y-components of the original vectors. 3 units1 unit 4 units 2 units + +=4 units 6 units

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One way to name the resultant vector is using components: 〈 4, 6 〉. 4 units 6 units

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You try it! What is the sum of the two vectors below?

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The resultant vector is shown below. It has components 〈 5, 6 〉. 5 units 6 units

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Another way of naming a vector is by giving its magnitude and direction. 5 units 6 units R = 〈 5, 6 〉

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The magnitude is the length of the vector. It can be found using the Pythagorean theorem. R = 〈 5, 6 〉 R x = 5 units R y = 6 units R = R x 2 + R y 2 = 5 2 + 6 2 = 7.810 units

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The direction is given as an angle. We can find it using trigonometry. R = 〈 5, 6 〉 R x = 5 units R y = 6 units R = 7.810 units = tan −1 = 50.19° 6 5

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Our resultant vector points 7.810 units 50.19° north of east because the angle was measured from due east. R = 〈 5, 6 〉 R = 7.810 units = 50.19° R = 7.810 units, 50.19° north of east

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If I measured the angle shown below, it would be called north of west because it is measured from due west.

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I could also describe this vector with a direction west of north if I measured the angle shown here.

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Try it yourself! How would you name these angles using compass points?

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It doesn’t matter which angle is smaller; it matters which axis you measure from! East of North South of East South of West North of West East of South West of South

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If I know the magnitude and direction of a vector, it is easy to calculate its x- and y-components.

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The vector shown below points 2.5 units 30° north of west. A = 2.5 units, 30° north of west 30°

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Its x-component is −2.5 cos 30° = −2.165 units. It is negative because it points to the left, along the −x axis. A = 2.5 units, 30° north of west 30° cos 30° = −A x A A x = −A cos 30° A x = −2.5 cos 30°

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Its y-component is 2.5 sin 30° = 1.25 units. It is positive because it points up, along the +y axis. A = 2.5 units, 30° north of west 30° sin 30° = AyAy A A y = A sin 30° A x = −2.165 units A y = 2.5 sin 30°

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You can always find A x = ±A cos and A y = ± A sin if you know the vector’s magnitude and angle with the x-axis. A = 2.5 units, 30° north of west 30° A y = 1.25 units A = 〈 −2.165 units, 1.25 units 〉 A x = −2.165 units

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The components of a vector can be + or − depending on its direction, but the magnitude is always positive. A = 2.5 units, 30° north of west 30° A = 〈 −2.165 units, 1.25 units 〉

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The end!

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