Vectors and Scalars Vector and scalar quantities Adding vectors geometrically Components Unit Vectors Dot Products Cross Products pps by C Gliniewicz

A scalar quantity is one which has magnitude only. Examples include mass, the number of planets, a student’s grades. A vector has both magnitude and direction associated with it. One first marks the paper with a direction such as north. To add vectors graphically, one draws the first vector as an arrow using some scale which fits on the paper and places an arrowhead in the direction it points. Then the vector which is to be added is drawn starting at the head of the first vector using the same scale. The sum of the two vectors is the line connecting the tail of the first vector to the head of the second vector. The length is determined using the same scale and the direction is determined using a protractor. To subtract a vector, one again draws the first vector and places an arrowhead in the proper direction. The second vector which is to be subtracted is then drawn from the head of first vector, but in the opposite direction. The answer is a vector drawn from the tail of the first to the head of the second. pps by C Gliniewicz

Vector equations are written similarly to regular equations, but an arrow above the symbols denotes the quantity as a vector. Associative, commutative and distributive rules also apply to vectors. Below are these rules. Vectors are different from number quantities since the direction is included. The numbers 5 and 3 can be added and there is only one result, 8. However, vectors with magnitudes 5 and 3 may be added to any value between 2 and 8 depending on the direction of the vectors. They add to 8 if they point in the same direction and they add to 2 if they point in opposite directions. If they are at right angles to one another, the magnitude of the resultant is and by changing the angle, all other values can be obtained. pps by C Gliniewicz

A vector can be resolved into its components using trigonometric functions or by carefully drawing the vectors. Components of a vector are two or three values which point along axes which are at right angles and whose origin is at the tail of the vector. Using the Pythagorean Theorem the magnitude of the vector can be obtained from the components. Graphically, one can carefully draw the vector onto a set of axes and draw the triangle and then measure the two components. Using trigonometry, and knowing the angle between the x axis and the vector, one can use the sine and cosine to determine the components. Recall that from math angles are measured counter-clockwise from the x axis. Compass directions are measured clockwise from north however. Be aware of the directions stated in any problem. pps by C Gliniewicz

A unit vector has a magnitude of exactly one and points in the direction of the axis. The letters i, j, k denote the directions of the x, y, and z axes. Note the “hat” over the unit vectors which is unique to those vectors. The quantities with the subscripts x, y, and z are scalar quantities. Combining them with the unit vectors gives a vector. To add the vectors a and b from above, one adds the coefficients of the I, j, and k unit vectors. pps by C Gliniewicz

When multiplying a vector by a scalar quantity, one needs only to multiply each term by the scalar. If the scalar were three, then each term is multiplied by three and the vector will be three times larger. There are two methods of multiplying a vector by another vector. There is the scalar product, also known as the dot product. There is also the vector product, also known as the cross product. The dot product is written as To find the dot product, one multiplies the i components and sums them with the product of the other components. If one of the components is zero, then the product will be zero. In the example above, if the b component in the second term were zero, the dot product would just be the first term. If the angle between the two vectors (when they are tail to tail) is known, the dot product can be calculated. The angle can be calculated if the components are known by setting the dot products equal to one another. The value between the absolute value signs is the magnitude of the vector determined by the Pythagorean Theorem. pps by C Gliniewicz

The cross product produces a vector for the answer and is written as The direction of the answer is determined by the right hand rule. First, one looks at the two vectors and uses the angle which is less than 180 degrees. Pointing the fingers of your right hand in the direction of the a vector and curling them toward the b vector, your thumb points at right angles to both a and b and in the direction of the answer. If the angle between the vectors (when they are tail to tail) is known, the cross product is If the angle, , between the two vectors is zero, then the component of one vector along the other is maximum and so is the dot product. The cosine function is one. If the angle is a right angle, the cosine is zero and the dot product is zero. The cross product can also be calculated using matrices. If the two vectors are parallel or antiparallel, the sine is zero and the cross product is zero. If the vectors are at right angles, the cross product is a maximum. pps by C Gliniewicz