Presented by: S. K. Pandey PGT Physics K. V. Khandwa Kinematics Vectors

Vector Addition Tip to tail method Parallelogram method 8 N 4 N 3 N Suppose 3 forces act on an object at the same time. F net is not 15 N because these forces aren’t working together. But they’re not completely opposing each either. So how do find F net ? The answer is to add the vectors... not their magnitudes, but the vectors themselves. There are two basic ways to add vectors w/ pictures:

Tip to Tail Method in-line examples Place the tail of one vector at the tip of the other. The vector sum (also called the resultant) is shown in red. It starts where the black vector began and goes to the tip of the blue one. In these cases, the vector sum represents the net force. You can only add or subtract magnitudes when the vectors are in-line! 16 N 20 N 4 N 20 N 16 N 12 N 9 N 12 N 21 N

Tip to Tail – 2 Vectors 5 m 2 m To add the red and blue displacement vectors first note: Vectors can only be added if they are of the same quantity—in this case, displacement. The magnitude of the resultant must be less than 7 m (5 + 2 = 7) and greater than 3 m (5 - 2 = 3). 5 m 2 m blue + red Interpretation: Walking 5 m in the direction of the blue vector and then 2 m in the direction of the red one is equivalent to walking in the direction of the black vector. The distance walked this way is the black vector’s magnitude. Place the vectors tip to tail and draw a vector from the tail of the first to the tip of the second.

Commutative Property blue + red red + blue As with scalars quantities and ordinary numbers, the order of addition is irrelevant with vectors. Note that the resultant (black vector) is the same magnitude and direction in each case.

Tip to Tail – 3 Vectors We can add 3 or more vectors by placing them tip to tail in any order, so long as they are of the same type (force, velocity, displacement, etc.). blue + green + red

Parallelogram Method This time we’ll add red & blue by placing the tails together and drawing a parallelogram with dotted lines. The resultant’s tail is at the same point as the other tails. It’s tip is at the intersection of the dotted lines. Note: Opposite sides of a parallelogram are congruent.

Comparison of Methods red + blue Tip to tail method Parallelogram method The resultant has the same magnitude and direction regardless of the method used.

Opposite of a Vector v - v If v is 17 m/s up and to the right, then -v is 17 m/s down and to the left. The directions are opposite; the magnitudes are the same.

Scalar Multiplication x -2x 3x3x Scalar multiplication means multiplying a vector by a real number, such as 8.6. The result is a parallel vector of a different length. If the scalar is positive, the direction doesn’t change. If it’s negative, the direction is exactly opposite. Blue is 3 times longer than red in the same direction. Black is half as long as red. Green is twice as long as red in the opposite direction. ½ x

Vector Subtraction red - blue blue - red Put vector tails together and complete the triangle, pointing to the vector that “comes first in the subtraction.” Why it works: In the first diagram, blue and black are tip to tail, so blue + black = red  red – blue = black. Note that red - blue is the opposite of blue - red.

Other Operations Vectors are not multiplied, at least not the way numbers are, but there are two types of vector products that will be explained later. –Cross product –Dot product –These products are different than scalar mult. There is no such thing as division of vectors –Vectors can be divided by scalars. –Dividing by a scalar is the same as multiplying by its reciprocal.

Comparison of Vectors 15 N 43 m km 27 m/s Which vector is bigger? The question of size here doesn’t make sense. It’s like asking, “What’s bigger, an hour or a gallon?” You can only compare vectors if they are of the same quantity. Here, red’s magnitude is greater than blue’s, since km = 56 m > 43 m, so red must be drawn longer than blue, but these are the only two we can compare.

Vector Components 150 N Horizontal component Vertical component A 150 N force is exerted up and to the right. This force can be thought of as two separate forces working together, one to the right, and the other up. These components are perpendicular to each other. Note that the vector sum of the components is the original vector (green + red = black). The components can also be drawn like this:

Dot Products First recall vector addition in component form:  x 1, y 1, z 1  x 2, y 2, z 2  + =  x1 + x2, y1 + y2, z1 + z2  x1 + x2, y1 + y2, z1 + z2  It’s just component-wise addition. Note that the sum of two vectors is a vector. For a dot product we do component-wise multiplication and add up the results:  x 1, y 1, z 1  x 2, y 2, z 2   = x 1 x 2 + y 1 y 2 + z 1 z 2 Note that the dot product of two vectors is a scalar! Ex:  -2, 3, 10  N  1, 6, -5  m = = -34 N m  Dot products are used to find the work done by a force applied over a distance, as we’ll see in the future.

Dot Product Properties The dot product of two vectors is a scalar. It can be proven that a  b = a b cos , where  is the angle between a and b. The dot product of perpendicular vectors is zero. The dot product of parallel vectors is simply the product of their magnitudes. A dot product is commutative: A dot product can be performed on two vectors of the same dimension, no matter how big the dimension. a  b = b  ab  a

Unit Vectors in 2-D The vector v =  -3, 4  indicates 3 units left and 4 units up, which is the sum of its components: v =  -3, 4  =  -3, 0  +  0, 4  Any vector can be written as the sum of its components. Let’s factor out what we can from each vector in the sum: v =  -3, 4  = -3  1, 0  + 4  0, 1  The vectors on the right side are each of magnitude one. For this reason they are called unit vectors. A shorthand for the unit vector  1, 0  is i. A shorthand for the unit vector  0, 1  is j. Thus, v =  -3, 4  = -3 i + 4 j

Unit Vectors in 3-D v =  7, -5, 9  =  7, 0, 0  +  0, -5, 0  +  0, 0, 9  One way to interpret the vector v =  7, -5, 9  is that it indicates 7 units east, 5 units south, and 9 units up. v can be written as the sum components as follows: = 7  1, 0, 0  - 5  0, 1, 0  + 9  0, 0, 1  = 7 i - 5 j + 9 k In 3-D we define these unit vectors: i =  1, 0, 0 , j =  0, 1, 0 , and k =  0, 0, 1  (continued on next slide)

Unit Vectors in 3-D (cont.) x y z 1 i j 1 k 1 The x-, y-, and z-axes are mutually perpendicular, as are i, j, and k. The yellow plane is the x-y plane. i and j are in this plane. Any point in space can be reached from the origin using a linear combination of these 3 unit vectors. Ex: P = (-1.8, -1.4, 1.2) so the vector P -1.8 i – 1.4 j k will extend from the origin to P.

Cross Products Let v 1 =  x 1, y 1, z 1  and v 2 =  x 2, y 2, z 2 . By definition, the cross product of these vectors (pronounced “v 1 cross v 2 ”) is given by the following determinant. v1  v2 =v1  v2 = x 1 y 1 z 1 x 2 y 2 z 2 i j k = (y 1 z 2 - y 2 z 1 ) i - (x 1 z 2 - x 2 z 1 ) j + (x 1 y 2 - x 2 y 1 ) k Note that the cross product of two vectors is another vector! Cross products are used a lot in physics, e.g., torque is a vector defined as the cross product of a displacement vector and a force vector. We’ll learn about torque in another unit.

a  ba  b a  b. Right hand rule  b a a  ba  b A quick way to determine the direction of a cross product is to use the right hand rule. To find a  b, place the knife edge of your right hand (pinky side) along a and curl your hand toward b, making a fist. Your thumb then points in the direction of It can be proven that the magnitude of is given by: a b sin  | a  b | =| a  b | = where  is the angle between a and b.

Dot Product vs. Cross Product 1. The dot product of two vectors is a scalar; the cross product is another vector (perpendicular to each of the original). 2. A dot product is commutative; a cross product is not. In fact, a  b = - b  a.- b  a.  x 1, y 1, z 1  x 2, y 2, z 2   = x 1 x 2 + y 1 y 2 + z 1 z 2 3. Dot product definition: Cross product definition: v1  v2 =v1  v2 = x 1 y 1 z 1 x 2 y 2 z 2 i j k 4. a  b = a b cos , and a b sin  | a  b | =| a  b | =