Consider Three Dimensions z y x a axax ayay azaz   xy Projection Polar Angle Azimuthal Angle

Unit Vectors Unit vectors are quantities that specify direction only. They have a magnitude of exactly one, and typically point in the x, y, or z directions. The i (or i ) unit vector points in the +x direction. The j (or j ) unit vector points in the +y direction. The k (or k ) unit vector points in the +z direction. ^ ^ ^

Unit Vectors z y x i j k

Instead of using magnitudes and directions, vectors can be represented by their components combined with their unit vectors. Example: displacement of 30 meters in the +x direction added to a displacement of 60 meters in the –y direction added to a displacement of 40 meters in the +z direction yields a displacement of:

Adding Vectors Using Unit Vectors Simply add all the i components together, all the j components together, and all the k components together. s = a + b where a = a x i + a y j b = b x i + b y j s = (a x + b x )i + (a y + b y )j s 2 = s x 2 + s y 2 tan  = s y / s x

Sample Problem A = 3.00 i j and B = i j Calculate C where C = A + B.

Sample Problem You move 10 meters north and 6 meters east. You then climb a 3 meter platform. What is your displacement vector? (Assume East is in the +x direction).

Sample Problem A = 3 i + j – 2 k and B = - i + 2 j Calculate │ A + B │.

Sample Problem A = 3.00 i j and B = i j Calculate D where D = A - B. Determine its magnitude and direction.

Multiplication of Vectors There are three ways that vectors can be multiplied. The resulting products are: –The product of a scalar and a vector –The scalar product of two vectors –The vector product of two vectors

The product of a scalar and a vector When you multiply a vector by a scalar, you get a vector that is parallel or anti-parallel to the original vector but may have a different magnitude and perhaps has different units. Applications –Unit vectors R = R x i + R y j + R z k –Inverse vectors –R = (-1)R –Momentump = mv –Electric force from electric field F = qE

The scalar product of two vectors (“dot product”) When you multiply a vector by a vector to get a scalar (dot product) you get a scalar whose magnitude depends upon how closely the vectors are aligned with each other. Applications –WorkW = F  d –PowerP = F  v –Magnetic FluxΦ B = B  A The quantities shown above are biggest when the vectors are completely aligned and there is a zero angle between them.

C = A  B C = AB cos  C = A x B x + A y B y + A z B z A B  The scalar product of two vectors (“dot product”)

The vector product of two vectors (“cross product”) When you multiply a vector by a vector to get a vector (cross product), you get a vector that is normal (perpendicular) to the plane established by the other two vectors Application –Torque  = r  F –Magnetic forceF = qv  B –Angular momentum l = r  p Cross products are biggest for vectors that are perfectly perpendicular.

C = A  B C = AB sin  with direction determined by Right Hand Rule A B  ijk C =A x A y A z B x B y B z The vector product of two vectors (“cross product”)

Sample Problem A: 70 at 45 o B: 35 at 79 o. Determine A ● B. Assume A and B are both in the x,y plane

Sample Problem A: 70 at 45 o B: 35 at 79 o. Determine A x B. Assume A and B are both in the x,y plane

Sample Problem A: 3 i – 2 j B: -6 i + 4 j Determine A ● B.

Sample Problem A: 3 i – 2 j B: -6 i + j Determine A x B.

Sample Problem A: 45 N North B: 15 m East Determine A ● B.

Sample Problem A: 45 N in North B: 15 m/s East Determine A x B.

Sample Problem A: 2 i – 4 j + 3 kB: - i + 2 j - 2k Determine A ● B.

Sample Problem A: 2 i – 4 j + 3 kB: - i + 2 j - 2k Determine A x B.

Sample Problem A baseball outfielder throws a long ball. The components of the position are x = (30 t)m and y = (10 t – 4.9t 2 )m a) Write vector expressions for the ball’s position, velocity, and acceleration as functions of time.

Sample Problem A particle undergoing constant acceleration changes from a velocity of 4i – 3j to a velocity of 5i + j in 4.0 seconds. What is the acceleration of the particle during this time period? What is its displacement during this time period?