Vectors

Vectors are physical quantities that have both a magnitude and a direction, as opposed to scalars, which need only a magnitude. Examples of vector quantities include displacement, velocity, acceleration, force, electric field, magnetic field, and momentum. Scalar quantities include distance, speed, temperature, volume, work, energy, and electric charge. Vectors are usually written with small arrows over the symbol or in boldface type. Mathematical operations with vectors are different than with ordinary numbers.

Vector Addition

Vectors can be added graphically be drawing to scale and placing the head of each successive vector at the tail of the preceding vector. The sum or resultant vector is a vector from the origin to the endpoint. Vector addition is commutative (a + b = b + a) and associative (a + b) + c = a + (b + c), so the order of addition makes no difference. When drawing vector addition, vectors may be mentally picked up and moved from their point of action as long as their length and spatial orientation is not changed.

Adding vectors algebraically can be done by using a coordinate system. Resolve vectors into components on the x and y axes (and z if necessary) and the sum of all the components gives the components of the vector sum. The actual vector magnitude and direction can be found if necessary. In physics we often use unit vectors to denote these vector components. These are vectors one unit long that point in the direction of the x, y, and z, axes labeled i, j, and k. So the x, y, and z components are multiplied by i, j, and k. These components are then added for several vectors and become the components of the vector sum in terms of unit vectors.  For example, if a + b = c, we would write the components with unit vectors like so:

a = a_xi + a_yj +a_zk

b = b_xi + b_yj +b_zk

and so  c = (a_x + b_x)i + (a_y + b_y)j + (a_z + b_z)k

 It is convenient to place the origin of the coordinate system in a convenient location and orientation with respect to the vectors in order to simplify the situation, but placing it elswhere doesn’t really change the result.

Vector Resolution

To resolve vectors into components, use right triangle geometry and trigonometry. For a vector a with magnitude of a acting at an angle q with the x axis, the x component a_x = a cosq, and the y component a_y = a sinq. Working in reverse, if the components are known  and

Vector Products

1. A scalar times a vector: results in a vector in the same direction as the original vector. Example: F = ma
2. The dot product (scalar product): Two vectors multiplied together produce a scalar whose magnitude is equal to the product of the magnitude of the two vectors times the cosine of the angle between them. It is called the dot product because it is always written with a dot to signify multiplication. The angle is measured between the vectors placed at a common origin.  To be correct this is actually the product of the magnitude of one vector times the scalar component of the other vector in the same direction as the first. If the angle between the vectors is 90^o, the dot product is zero. Example: , the scalar quantity work equals the dot product of two vectors, force and displacement. The order of multiplication makes no difference in the dot product. In unit vector notation, aּb = (a_xi + a_yj + a_zk) ּ (b_xi + b_yj + b_zk) which obeys the distributive law. Since the angle between the three unit vectors is 90^o, any terms created during multiplication containing two different unit vectors will equal zero.
3. The cross product (vector product): This is the product of two vectors that produces and third vector perpendicular to both of the original vectors. It is written with an x to show multiplication, hence the name cross product. The magnitude of the cross product equals the product of the magnitudes of the two vectors times the sin of the smaller of the two angles that can be measured between them. Once again the vectors must have the same origin. We write:

a x b = c and  c = ab sinq  The direction of the new vector can be found using the right-hand rule: with the two vectors placed with their tails at the same origin, align rhe fingers of your right hand with vector a, the first vector in the cross product so that you could sweep a into b through the smaller angle between them. Extend your thumb upwards and it will point in the direction of the product, vector c. If the order of multiplication is reversed, the product will be in the opposite direction, so b x a = - a x b. Also, since the product depends on the sin of the angle, it will be a maximum when the two vectors are perpendicular and zero when they are parallel or anti-parallel. This is important to remember when doing vector products in unit-vector notation. a x b = (a_xi + a_yj + a_zk) x (b_xi + b_yj + b_zk) Once again, the distributive law applies, but any terms having a unit vector multiplied by itself will be equal to zero. Products involving two different unit vectors will yield the third unit vector, either positive or negative, depending on the order of the cross product. So i x j = k but j x i = -k and so on. An example of a cross product we encountered during first year physics is torque, produced by a force acting at a radial distance from the point of rotation, usually written