Linear Algebra

246 Chapter Three. Maps Between SpacesFor the wind we use a vector of length 15 that points toward the northeast.( √ )15 1/2⃗v =15 √ 1/2The car can only be affected by the part of the wind blowing in the east-westdirection — the part of ⃗v in the direction of the x-axis is this (the picture hasthe same perspective as the railroad car picture above).northeast( √ )15 1/2⃗p =0So the car will reach a velocity of 15 √ 1/2 miles per hour toward the east.Thus, another way to think of the picture that precedes the definition is thatit shows ⃗v as decomposed into two parts, the part with the line (here, the partwith the tracks, ⃗p), and the part that is orthogonal to the line (shown here lyingon the north-south axis). These two are “not interacting” or “independent”, inthe sense that the east-west car is not at all affected by the north-south partof the wind (see Exercise 11). So the orthogonal projection of ⃗v into the linespanned by ⃗s can be thought of as the part of ⃗v that lies in the direction of ⃗s.Finally, another useful way to think of the orthogonal projection is to havethe person stand not on the line, but on the vector that is to be projected to theline. This person has a rope over the line and pulls it tight, naturally makingthe rope orthogonal to the line.That is, we can think of the projection ⃗p as being the vector in the line that isclosest to ⃗v (see Exercise 17).1.6 Example A submarine is tracking a ship moving along the line y = 3x+2.Torpedo range is one-half mile. Can the sub stay where it is, at the origin onthe chart below, or must it move to reach a place where the ship will pass withinrange?northeast