a) Find a unit vector that has the same direction as the given

vector. 8**i** ? **j** +

4**k**

**b)** Find a vector that

has the same direction as <-2,4,2> but has length 6

c) If **v**

lies in the first quadrant and makes an angle *?*/3 with the

positive *x*-axis and |**v**| = 8, find

**v** in component form.

.

Tony SparkEnlightened

(a) Consider the following =8i-j+4k vector a. The unit vector that has the same direction of the vector a is, 2 u The magnitude of a =8i -j+4k is, +4 = 641+16 = V81 Therefore, the unit vector that has same direction of the vector is 2 8i-j+4k C (b) Consider the following vector b-2,4,2)2i+ 4j+2k The magnitude of this vector is b -2i+4j+2k |(-2) +42 +2 4+16+4 24 = 6×4 -26 The unit vector that has same direction of the vector is b u -2i +4j2k 216 Thus, the unit vector with length 6 units is, -2i+42k 216 6u = 6 =V6 2i+4j+2k (-2i +4j +2k) 2 (2(-i+2j+k)) 6(-i+2j+k) (c) Consider the following diagram: 3 = v] = 8 and e From the graph, the magnitude of the vector r then 3 The object is to find the component vector for v. From the graph, the component vectors for vare given as, =rcos e and b = rsin 0 8 and e Then, plug into a rcos and b = rsin 8 3 r = 8 cos a = 4 b 8 sin 3 V3 2 =413 Thus, the component vector vis, (4,43 n