Key Concepts for the Lecture of 1Jul03
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We use vectors to represent physical quantities that possess both magnitude and direction in space. A quantity with magnitude only is called a scalar.
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We graphically represent a vector by an arrow. The length of the arrow represents the magnitude; the direction the arrow points represents the direction.
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Vectors are notated by placing an arrow above the letter standing for the vector. The magnitude of a vector is properly expressed by enclosing it between vertical strokes, but we often just leave the arrow off.
vector
3
magnitude | A | ( or just A )
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Parallel Transportation. A vector has only magnitude and direction; its position is irrelevant to its value. You may therefor move the arrow representing a vector around on the page as long as you keep its length constant and don’t change its direction.
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Unit Vectors. A unit vector is a vector with maginitude (length) unity. It therefor conveys only direction. Unit vectors are notated with a hat (circumflex) over the letter representing the vector.
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Basis Vectors. The directions of the three axes in a conventional Cartesian coordinate system are given by three dimensionless unit vectors that lie along the axes. These vectors are called the orthogonal basis vectors for the coordinate system.
i
ˆ lies along the x axis j
ˆ lies along the y axis
ˆ axis
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Scalar Multiplication. If you multiply a vector by a scalar you get a new vector with the same direction but a magnitude scaled by the multiplication with the scalar.
.
3
B
3 3
| B |
.
| A |
Multiplying a vector by -1 reverses its direction.
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Addition. The sum of two vectors is a third vector called the resultant. There are two equivalent methods of graphical addition, the tip-to-tail method and the parallelogram
method. Add two vectors by arranging them (through parallel transportation) so the tip of one touches the tail of the other. The resultant is the vector drawn from the tail of the first to the tip of the second. Equivalently, place the vectors with their tails together, and complete the construction of the parallelogram for which the vectors form two sides. The resultant is the diagonal with its tail at the tail of the two addends and its tip at the opposite vertex of the

parallelogram.
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Subtraction. To subtract A - A = D , place the A and A vectors with their tails together and
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Scalar Product ( also known as the Inner Product or Dot Product and not to be confused with
Scalar Multiplication). The dot product multiplies two vectors to produce a scalar:
3
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B A B cos
The dot product represents the projection of A onto B ( ie A cos ) times the magnitude of B.
This is the same as the projection of B onto A (ie B cos ) times the magnitude of A.
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Composition of Vectors. Any vector can be expressed as a sum of three vectors lying along the coordinate axes:
3
3
3 x y z
3 3 x where
3 y
3 z the projection of A onto the x axis the projection of A onto the y axis the projection of A onto the z axis
Consequently, any vector may be expressed algebraically as
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A x i
ˆ
A y j
ˆ
A z k
ˆ
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In two dimensions (x, y), if vector A has a magnitude A and makes an angle with the x-axis,
3
A cos i
ˆ
A sin j
ˆ
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Because of the Pythagorean Theorem, the magnitude of a vector is
| A | A x
2
A y
2
A z
2
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Addition.
3 3
( A x
B x
) ˆ ( A y
B y
) ˆ ( A z
B z

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Dot Product.
3
#
3
( A x
B x
) ( A y
B y
) ( A z
B z
)
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Derivative.
d A dt dA x dt i
ˆ dA y dt j
ˆ dA z dt k
ˆ
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Position Vector. Note that the components of the position vector are just the coordinates of the position: r 3 x ˆ y ˆ k
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Displacement Vector. The components of the displacement vector are just the differences of the coordinates: s 3
21 r 3
2 r 3
1
( x
2 x
1
) ˆ ( y
2 y
1
) ˆ ( z
2 z
1 s
21
( x
2 x
1
)
2
( y
2 y
1
)
2
( z
2 z
1
)
2
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Velocity Vector. The components of the velocity vector are the derivatives of the individual coordinates: d r 3 dt dx dt i
ˆ dy dt j
ˆ dz dt k
ˆ
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Caution. Coordinate axes are ruled off in length units (typically meters). The components of the position vector are lengths along the axes. But when you draw any other type of vector (a velocity vector for example) on a space diagram, and project it onto the axes to find its components, the lengths of the components are not in the units of the axes. You may scale the components of a non-position vectors as you wish. The basis unit vectors are dimensionless.