Physics Ch.3 Vectors

3.1 – 3.2 Scalars & Vectors Scalars -A quantity that states only an amount (magnitude only). Examples – Speed, distance, time, mass temperature. They have units but no direction Vectors - A quantity specified by both magnitude and direction Examples – displacement, velocity, acceleration, and force. Represented by arrows whose length represents the magnitude and point in the direction of the vector Meaning is defined by its length and direction, not by its starting point.

3.3 Polar Notation Defining a vector by its angle and magnitude. magnitude and direction of the vector are stated separately. angle is typically measured from the positive side of the x axis to the vector. Angles can be positive or negative. A positive angle indicates a counterclockwise direction, a negative angle a clockwise direction. Use degrees to specify angles unless we specifically note that we are using radians. (Radians do prove essential at times.) Polar Notation

3.4 Vector Components and Rectangular Notation Rectangular notation: Defining a vector by its components. The x component of a vector indicates its extent in the horizontal dimension the y component its extent in the vertical dimension. The x and y components of a vector are written inside parentheses. A vector that extends a units along the x axis and b units along the y axis is written as (a, b) The components of vectors are scalars with the direction indicated by their sign: x components point right (positive) or left (negative), y components point up (positive) or down (negative). Rectangular Notation

3.5 Adding and Subtracting Vectors Graphically To add two vectors, you: Place the tail of the second vector at the head of the first vector. (The order of addition does not matter, so you can place the tail of the first vector at the head of the second as well.) Draw a vector between the tail end of the first vector and the head of the second vector. This vector represents the sum of two vectors. Vector addition

3.6 Adding and subtracting vectors by components You can combine vectors graphically, but it may be more precise to add up their components. Break the vector into its components and add each component independently. All types of vectors can be added or subtracted. You can add two velocity vectors, two acceleration vectors, two force vectors and so on. Vector subtraction works similarly to addition when you use components. For example, (5, 3) minus (2, 1) equals 5 minus 2, and 3 minus 1; the result is the vector (3, 2). Component addition

3.7 Interactive checkpoint: vector addition

3.8 Interactive checkpoint: a jogger Vector Components II

3.9 Multiplying rectangular vectors by a scalar To multiply a vector by a scalar, multiply each component of the vector by the scalar. An airplane travels at a constant velocity represented by the vector (40, 10) m/s. You know its current position and want to know where it will be if it travels for two seconds. In this example, (2 s)(40, 10) m/s = (80, 20) m. This is the plane’s displacement vector after two seconds of travel. If you wanted the opposite of this vector, you would multiply by negative one. The result in this case would be (−40, −10) m/s, representing travel at the same speed, but in the opposite direction Multiplying Vectors and Scalars

3.10 Multiplying polar vectors by a scalar Multiplying a vector represented in polar notation by a positive scalar requires only one multiplication operation: Multiply the magnitude of the vector by the scalar. The angle is unchanged. If you multiply a vector by a negative scalar, multiply its magnitude by the absolute value of the scalar (that is, ignore the negative sign). Then change the direction of the vector by 180° so that it points in the opposite direction. In polar notation, since the magnitude is always positive, you add 180° to the vector's angle to take its opposite. The result of multiplying (50 km, 30°) by negative three is (150 km, 210°). Polar vectors

3.11 Converting vectors from polar to rectangular notation Converting the vectors above requires some trigonometry basics, namely sines and cosines. Treat the magnitude of a vector as the hypotenuse of a right triangle, with the x component as its horizontal leg and the y component as its vertical leg. The x and y components can be positive or negative. For instance, the x component will be negative when the cosine is negative, which it is for angles between 90° and 270°. The y component will be positive when the sine is positive (between 0° and 180°, the vector has an upward y component) and negative when the sine is negative (between 180° and 360°, the vector has a downward y component). it is a good practice to compare directions and the signs of the components. Converting Vectors

3.12 Converting vectors from rectangular to polar notation To convert from rectangular to polar coordinates use trigonometry . The x and y components represent the legs of a triangle. You need to determine the length of the hypotenuse and the angle the hypotenuse makes with the positive x axis Now determine the angle, which is represented as θ. The tangent function relates the base and height of a right triangle to the angle between the hypotenuse and the base. The angle θ is the arctangent of the ratio of the two legs of the triangle. Converting Vectors II

3.13 Sample problem: driving in the desert Driving Problem

3.14 Sample problem: looking ahead to forces Forces Problem

3.15 Interactive checkpoint: a bum steer Bum Steer Problem

3.16 Unit Vectors A vector can be described by its components in rectangular notation, as with (20, 30, 40). This describes a vector that extends 20 units in the x direction, 30 units in the y direction, and 40 units in the z direction. Another form of notation called unit vector notation has much in common with rectangular notation. In this notation, the vector (a, b, c) is written as ai + bj + ck. The unit vectors i, j and k have lengths equal to one and point along the x, y and z dimensions respectively. So, ai is the product of the scalar a and the unit vector i. If a is positive, the result is a vector of magnitude a pointing in the positive x direction. If a is negative, the vector ai points in the negative x direction, with magnitude equal to the absolute value of a. Unit vectors are dimensionless; there are no units associated with them. The product ai will have the same units as a. For example, (3 m/s)i + (4 m/s)j represents a velocity vector of three meters per second in the x direction and four meters per second in the y direction. This can also be written as 3i + 4j m/s. Unit Vectors

3.17 Interactive summary problem: back to base Dock Problem