A graphic artist is creating a new logo for her company’s website. In the graphics program she is using, each pixel in an image file has coordinates (?x?, y?), where the origin (0, 0) is at the upper left corner of the image, the +?x?-axis points to the right, and the +?y?-axis points down. Distances are measured in pixels. (a) The artist draws a line from the pixel location (10, 20) to the location (210, 200). She wishes to draw a second line that starts at (10, 20), is 250 pixels long, and is at an angle of 30° measured clockwise from the first line. At which pixel location should this second line end? Give your answer to the nearest pixel. (b) The artist now draws an arrow that connects the lower right end of the first line to the lower right end of the second line. Find the length and direction of this arrow. Draw a diagram showing all three lines.
Solution 77P a) The graphical representation is given below. The 1st line is given here. Then she made the 2nd line as follows. The length of the vector say R is 250. The angle with the horizontal is 30 . 0 So the horizontal component length will be, 0 R x Rcos = 250 × cos 30 = 250 × 0.866 = 216(approximately) The vertical component length will be, R y R sin = 250 × sin 30 = 250 × 0.5 = 125. The initial position was (x 1y 1 = (10,20). The final position will be, (x2,y2) = (x1+ 216, y 1 125) = (10 + 216, 20 + 125) = (226, 145). So the second line would end at the point of (226, 145). b) There is one more line which connects the lower right of the 1st and 2nd line. Here the 1st line is represented as vector A. 2nd line as vector R. The joining line as vector B. We know from vector addition that, A + B = R B = R A = R +A 2RA cos ----------------(1) And R value we already know , which is 250. Length of A vector is, (210 10) + (200 20) =269 To find the angle of A vector with the horizontal, The x component of A is 200 and the y component is 180. So, tan = 180/200 = 0.9 1 0 = tan (0.9) = 42 . Now by putting all the values in equation (1), 2 2 B = R A = + A 2RA cos = 50 + 269 2 × 250 × 269 × cos 42 0 = 62500 + 72361 99953 = 34908 = 187 CONCLUSION: So, the magnitude of this vector will be 187 pixels. The direction is towards the negative y direction.