2. 2
Vectors - revision
A vector is a physical quantity with magnitude and direction.
Vectors can be used to represent many physical quantities that have a magnitude and
direction, like forces.
Vectors may be represented as arrows where the length of the arrow indicates the
magnitude and the arrowhead indicates the direction of the vector.
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3. 3
Vectors on the Cartesian plane
We can represent vectors on the Cartesian plane. The vectors can be placed anywhere
on the Cartesian plane as long as the magnitude and direction of the vector is correctly
indicated.
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4. 4
Vectors on the Cartesian plane continued
Vectors can also be drawn at an angle to one of the axes. When we do this we
specify the angle as acting anti-clockwise from the positive x-axis.
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5. 5
The resultant vector
The resultant of a number of vectors is the single vector whose effect is the same as the
individual vectors acting together.
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6. 6
Sketching the resultant
We can sketch vectors in two dimensions using the tail-to-head method. In this method the
tail of one vector is placed at the head of the other vector.
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7. 7
Sketching the resultant continued
We can also first find the resultant in the x-direction and then find the resultant in the
y-direction before finding the final resultant.
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8. 8
Tail-to-tail method of sketching the resultant
We can use the tail-to-tail method to sketch the resultant. This is illustrated below.
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9. 9
Closed vector diagrams
A closed vector diagram is a set of vectors drawn on the Cartesian using the tail-to-head
method and that has a resultant with a magnitude of zero. This means that if the first vector
starts at the origin the last vector drawn must end at the origin. The vectors form a closed
polygon, no matter how many of them are drawn. Here are a few examples of closed vector
diagrams:
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10. 10
Finding the resultant
Using the algebraic techniques of vector
addition from grade 10 and Pythagoras'
theorem we can find the magnitude of the
resultant of vectors in two dimensions.
We first find the resultant in the x-direction
and then find the resultant in the y-direction.
Finally we note that when we draw these
two vectors head-to-tail we get a right
angled triangle that has the resultant of the
vectors as the hypotenuse.
The magnitude of the resultant can be
calculated algebraically or measured
graphically from a scale diagram.
The direction can be measured from a scale
diagram or calculated using trigonometry.
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11. 11
Components of vectors
We can resolve any vector into components. This can be done algebraically or
graphically. We can use:
and
R x =R cos R y =R sin
to calculate the magnitude of the x- and y-components.
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12. 12
Components of vectors continued
We can extend this to vectors in two dimensions.
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13. 13
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