Math 233 Vectors! Version: 8-29-05 Vector Factoids: Know The Difference! Scalar: a quantity represented by a real number Vector: a quantity represented by both a direction and a magnitude Displacement Vector: measures the displacement between two points A=(2, 3) and B=(5,2), then AB 5 2, 2 3 3, 1 Always remember: Head – Tail or To – From Magnitude: the magnitude of a vector a, b, c is: a 2 b2 c 2 Unit Vector: a vector whose magnitude is 1 (so it measures direction, or relative slope only!) Notation: b b 4,2 4i 2 ˆ where iˆ and ˆ are unit vectors in the positive x- ˆ j j and y- axis directions. More Notation: unit vectors are frequently given “hats” (or carets) to denote their unique property ( iˆ ). Equality: Two vectors are equal if they have the same length and direction Some Practice: 1. Find the displacement from the point P1 to P2 for P 4,6 1 P2 7,10 : 2. Given the vectors u and v shown, sketch the vectors 2u , u v , and u v . v u 3. Two adjacent sides of a regular hexagon are given as the vectors u and v . Label the remaining sides in terms of u and v (for example, u or u v ). v u 4. Given the vector v 2,4 . a) What is the magnitude of v (noted as v or sometimes v )? (Hint: draw a picture and then use the Pythagorean Theorem) b) Find a unit vector in the direction of v . (Hint: the magnitude of a unit vector is 1… what was the magnitude of v ?) 5. A vector has length 8 and makes a 30-degree angle with the x-axis. What are the values of the iˆ and ˆ components? j ˆ 6. The velocity of a truck is determined to be 9i 11 ˆ . If another truck is traveling twice as j fast in the opposite direction, what is the velocity vector of the second truck? What is the speed of the second truck (note: speed is a scalar, velocity is a vector)? 7. Super Bonus Problem: An airplane is flying at an airspeed of 600 km/hr in a cross-wind that is blowing from the northeast at a speed of 50 km/hr. In what direction should the plane head to end up going due east? What will the groundspeed end up being (this is the speed in the resultant, or due east, direction)?
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