Notes: Vectors and Scalars

Transcription

1 A particle moving along a straight line can move in only two directions and we can specify which directions with a plus or negative sign. For a particle moving in three dimensions; however, a plus sign or minus sign is no longer enough to indicate a direction, we must use a vector. What is a vector? An object that has both magnitude and direction. Ok, so what is magnitude? Mathematically, it is the absolute value. Really it is the size or length of something. If you say your drove 4 km, it doesn t matter if it was +4 km or 4 km, the magnitude is 4 km. You drove 4 km. What is direction? If on a single axis, we say that from the objects original position to the objects final position, can be represented by a plus sign or a minus sign. So what about when we are not talking about 1-D, but two or three dimensions? We use terminology such as North, South, East, West, Northeast, Northwest, Southeast, and Southwest. However, mathematically, it easier to describe it with angles such as 30 above the x-axis. What are examples of vectors? Displacement, velocity, acceleration, force Obviously, not all substance are vectors e.g. not all substances have direction, so what are substances without a direction called? Scalars. Scalar substances without direction. What are some examples of some substances with only magnitude and no direction? Temperature, pressure, energy, mass, and time.

2 (COMPONENTS OF A VECTOR) A component of a vector is the projection of the vector on an axis; that is, the part of the vector that translates to three-dimensional space. The process of finding the components of a vectors is called resolving the vector. So as in the image, we have a vector on an x-y plane: We can break it into component by: Basically, we have a component of the vector d on the x-axis that is called d x, and a component of it on the y-axis that is called d y. Looking at the diagram above, we see that essentially the vector d and its components form a triangle. This allows us to use trigonometry to create some basic formulas for describing a vector. Recall that:

3 cosθ = sinθ = adjacent hypotenuse opposite hypotenuse tanθ = opposite adjacent We can find d x (that is, the amount of vector d on the x-axis) by using cosθ cosθ = d x d This can be better written as the formula for d x as: d x = dcosθ Similarly, we can find d y (that is, the amount of vector d on the y-axis) by using sinθ sinθ = d y d This can be better written as the formula for d y as: d y = dsinθ So, the two above highlighted equations will give you the components of a vector.

4 However, what if we know the components of a vector, but not the magnitude of the vector or direction. Look back at the graph above, we have a triangle. Our magnitude is just the distance of the hypotenuse. So what equation can we use to find the hypotenuse? The Pythagorean theorem. So to find the magnitude of a vector, we use this formula: d = d x 2 + d y 2 As we already know by now, a vector has both magnitude and direction. We just found the formula for finding the magnitude, now how do we find the direction (angle)? We use tanθ to find the angle of a vector: tanθ = d y d x (UNIT-VECTOR NOTATION) Unit vector a vector that has a magnitude of exactly 1 and points in a particular direction. It lacks both dimension and unit. Its sole purpose is to point that is, to specify direction. The unit vectors in the positive direction of the x, y, and z axes are labeled i, j, and k. That is: i = x axis or x direction. j = y axis or y direction. k = z axis or z direction.

5 Unit vectors are very useful for expressing other vectors for instance: Where i and j are the vector components, and a x and a y are the scalar components of vector a. (ADDING VECTORS BY COMPONENTS) To add vectors, we can add their components axis by axis. Take r = a + b We can rewrite this as: r = (a x + a y + a ) z + (b x + b y + b ) z

6 r = (a x + b x )i + (a y + b y )j + (a z + b )k z r = r i x + r j y + r k z Where, r x = a x + b x r y = a y + b y r z = a z + b z Product of Vectors: - Let i, j, and k be unit vectors in the x, y, and z directions. Then: i i = j j = k k = 1 and i j = j k = k i = 0 Meaning that only vectors in the same direction can be multiplied together whereas if they are in opposite direction than they cancel out. (ADDING VECTORS GEOMETRICALLY) Resultant vector (or vector sum) the sum of two vectors that create another vector. To add two-dimensional vectors together geometrically: (1) Sketch a vector a to some convenient scale and at the proper angle. (2) Sketch a vector b to the same scale, with its tail at the head of vectors a, again at the proper angle (3) The vector sum s is the vector that extends from the tail of a to the head of b.

7 Many of the same laws that you encountered in algebra, apply in the same way to vectors. Commutative law states that the order of addition does not matter. Adding vector a to vector b gives to same result as adding vector b to vector a. Associative law is an extension of the commutative law to more than two vectors by stating that the order of addition does not matter for vectors a, b, and c. Vector Subtraction states that if you have a vector b than it has the same magnitude as vector b except it is in the opposite direction. NOTE: Although we have used displacement vectors here, the rules for addition and subtraction hold for vectors of all kinds, whether they represent velocities, accelerations, or any other vector quantity. (MULTIPLYING VECTORS) Three ways in which vectors can be multiplied: 1. Multiplying a Vector by a Scalar 2. Dot Product (Multiplying a Vector by a Vector) 3. Cross Product (Multiplying a Vector by a Vector) (1) Multiplying a Vector by a Scalar: If we multiply a vector a by a scalar s, we get a new vector. The new vectors magnitude is the product of the magnitude of vector a and the absolute value of

8 scalar s. The new vectors direction is the direction of vector a if scalar s is positive unless scalar s is negative then the new vectors direction is negative. Let s be the scalar and let a be a vector. Let d be the vector when the scalar and the vector a are multiplied together. (2) The Scalar Product (Dot Product): d = sa The Scalar Product that is most often called the dot product is one of the ways to multiply a vector by another vector which is to say to multiply a quantity that has a magnitude and a direction (a )by another quantity that has a magnitude and a direction ( b ). Where, θ is the angle between the two vectors. IMPORTANT NOTE: There are actually two such angles when two vectors are multiplied together. θ and (360 -θ) Either angle can be used in the dot product because their cosines are the same but this is not true for the cross product or anytime you have the sine of something. Dot Product produces a scalar product and not another vector i.e. when two vectors or multiplied together by the dot product, they produce a quantity with only a magnitude and not a direction. A dot product can be regarded as the product of two quantities: 1. The magnitude of one of the vectors. 2. The scalar component of the second vector along the first vector.

9 (3) The Vector Product (Cross Product): The Vector Product is commonly called the Cross Product. It is when you multiply two vectors together to produce another vector. Note that the word multiply is used loosely because it is different than how you typically accustomed to multiplying.

10 Essentially it is when a vector (a ) encounters another vector (b ) to create a new vector (c ) that has both a magnitude and a direction. That is: c = a x b = absin i j k i j k c = a x b = a x a y a z = a x a y a z b x b y b z b x b y b z i a x j a y b x b y = (a y b z a z b y )i + (a z b x a x b z )j + (a x b y a y b x )k is the small of the two angles between vector (a ) and vector (b ); this is because sin and sin(360 - ) differ in algebraic sign. The direction of vector c (which is the vector created from crossing vector a and vector b ) is perpendicular to the plane that contains vector a and vector b. EXAMPLE PROBLEM 1: A small airplane leaves an airport on an overcast day and is sighted 215 km away, in a direction making an angle of 22 East of due North. How far east is the airplane from the airport when sighted? How far north is the airplane from the airport when sighted? STEP 1: The first part to solving physics problems and any work problem is to pick out what relevant and irrelevant data you have in a problem. So ask yourself, what do we know so far? We have a number that 215 km, but what is that number? It is the magnitude of the displacement vector (look at graph). So, we can say that d = 215 km.

11 We also have an angle that is 22 east of due north. What does that mean? Well if you recall a unit circle where the y-axis is 90 and is pointing straight up, the same direction as North, we can call that our north direction. Similarly, east would be heading to the right. So basically the is 22 east of due north says that we start at north and we head east. So we are 22 degrees off the y-axis. So as you can see, we end up with an angle that is θ = = 68. Now we break the vector down into its components.

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