Here are two vectors: a = (2.0 m) i - (2.0 m) j and b = (4.0 m) i + (8.0 m) j. (
ID: 1952695 • Letter: H
Question
Here are two vectors:a = (2.0 m) i - (2.0 m) j and b = (4.0 m) i + (8.0 m) j.
(a) Find the magnitude of a.
m
Find the direction of a.
°
b) Find the magnitude of b.
m
Find the direction of b.
°
(c) Find the magnitude of a + b.
m
Find the direction of a + b.
°
(d) Find the magnitude of b - a.
m
Find the direction of b - a.
°
(e) Find the magnitude of a - b.
m
Find the direction of a - b.
°
How do the orientations of the last two (d and e) compare?
AThe angle between the two vectors is 45°.
BThe angle between the two vectors is 30°.
C The angle between the two vectors is 0°.
D The angle between the two vectors is 90°.
EThe angle between the two vectors is 180°.
F The angle between the two vectors is 60°.
Explanation / Answer
Given that, The first vector, a = (2.0 m)i^-(2.0 m)j^ The second vector, b = (4.0 m)i^+(8.0 m)j^____________________________________________ a) The magnitude of the vector 'a' is given by |a| = (2.0 m)2+(-2.0 m)2 = 2.828 m = 2.83 m The direction of the vector 'a' is tan = -2.0 m/2.0 m = tan-1(-1.0) = -45o (or) 135o ____________________________________________ ____________________________________________ b) The magnitude of the vector 'b' is given by |b| = (4.0 m)2+(8.0 m)2 = 8.944 m = 8.95 m The direction of the vector 'b' is tan = 8.0 m/4.0 m = tan-1(2.0) = 63.43o ______________________________________________ ______________________________________________ c) The sum of the two vectors is given by a+b = [(2.0 m)i^-(2.0 m)j^]+[(4.0 m)i^+(8.0 m)j^] = (6.0 m)i^+(6.0 m)j^ The magnitude of the vector 'a+b' is given by |a+b| = (6.0 m)2+(6.0 m)2 = 8.485 m = 8.49 m The direction of the vector 'a+b' is tan = 6.0 m/6.0 m = tan-1(1.0) = 45o ______________________________________________ ______________________________________________ d) The difference between vector b and vector a is b-a = [(4.0 m)i^+(8.0 m)j^]-[(2.0 m)i^-(2.0 m)j^] = (2.0 m)i^+(10.0 m)j^ The magnitude of the vector 'b-a' is given by |b-a| = (2.0 m)2+(10.0 m)2 = 10.198 m = 10.20 m The direction of the vector 'b-a' is tan = 10.0 m/2.0 m = tan-1(5.0) = 78.69o _________________________________________________ _________________________________________________ e) The difference between vector a and vector b is a-b = [(2.0 m)i^-(2.0 m)j^]-[(4.0 m)i^+(8.0 m)j^] = (-2.0 m)i^-(10.0 m)j^ The magnitude of the vector 'a-b' is given by |a-b| = (-2.0 m)2+(-10.0 m)2 = 10.198 m = 10.20 m The direction of the vector 'a-b' is tan = -10.0 m/-2.0 m = tan-1(5.0) = 78.69o (or) 258.69o __________________________________________________ __________________________________________________ f) The angle between the two vectors is 180°.
= 2.828 m = 2.83 m The direction of the vector 'a' is tan = -2.0 m/2.0 m = tan-1(-1.0) = -45o (or) 135o ____________________________________________ ____________________________________________ b) The magnitude of the vector 'b' is given by |b| = (4.0 m)2+(8.0 m)2 = 8.944 m = 8.95 m The direction of the vector 'b' is tan = 8.0 m/4.0 m = tan-1(2.0) = 63.43o ______________________________________________ ______________________________________________ c) The sum of the two vectors is given by a+b = [(2.0 m)i^-(2.0 m)j^]+[(4.0 m)i^+(8.0 m)j^] = (6.0 m)i^+(6.0 m)j^ The magnitude of the vector 'a+b' is given by |a+b| = (6.0 m)2+(6.0 m)2 = 8.485 m = 8.49 m The direction of the vector 'a+b' is tan = 6.0 m/6.0 m = tan-1(1.0) = 45o ______________________________________________ ______________________________________________ d) The difference between vector b and vector a is b-a = [(4.0 m)i^+(8.0 m)j^]-[(2.0 m)i^-(2.0 m)j^] = (2.0 m)i^+(10.0 m)j^ The magnitude of the vector 'b-a' is given by |b-a| = (2.0 m)2+(10.0 m)2 = 10.198 m = 10.20 m The direction of the vector 'b-a' is tan = 10.0 m/2.0 m = tan-1(5.0) = 78.69o _________________________________________________ _________________________________________________ e) The difference between vector a and vector b is a-b = [(2.0 m)i^-(2.0 m)j^]-[(4.0 m)i^+(8.0 m)j^] = (-2.0 m)i^-(10.0 m)j^ The magnitude of the vector 'a-b' is given by |a-b| = (-2.0 m)2+(-10.0 m)2 = 10.198 m = 10.20 m The direction of the vector 'a-b' is tan = -10.0 m/-2.0 m = tan-1(5.0) = 78.69o (or) 258.69o __________________________________________________ __________________________________________________ f) The angle between the two vectors is 180°.
|b| = (4.0 m)2+(8.0 m)2 = 8.944 m = 8.95 m The direction of the vector 'b' is tan = 8.0 m/4.0 m = tan-1(2.0) = 63.43o ______________________________________________ ______________________________________________ c) The sum of the two vectors is given by a+b = [(2.0 m)i^-(2.0 m)j^]+[(4.0 m)i^+(8.0 m)j^] = (6.0 m)i^+(6.0 m)j^ The magnitude of the vector 'a+b' is given by |a+b| = (6.0 m)2+(6.0 m)2 = 8.485 m = 8.49 m The direction of the vector 'a+b' is tan = 6.0 m/6.0 m = tan-1(1.0) = 45o ______________________________________________ ______________________________________________ d) The difference between vector b and vector a is b-a = [(4.0 m)i^+(8.0 m)j^]-[(2.0 m)i^-(2.0 m)j^] = (2.0 m)i^+(10.0 m)j^ The magnitude of the vector 'b-a' is given by |b-a| = (2.0 m)2+(10.0 m)2 = 10.198 m = 10.20 m The direction of the vector 'b-a' is tan = 10.0 m/2.0 m = tan-1(5.0) = 78.69o _________________________________________________ _________________________________________________ e) The difference between vector a and vector b is a-b = [(2.0 m)i^-(2.0 m)j^]-[(4.0 m)i^+(8.0 m)j^] = (-2.0 m)i^-(10.0 m)j^ The magnitude of the vector 'a-b' is given by |a-b| = (-2.0 m)2+(-10.0 m)2 = 10.198 m = 10.20 m The direction of the vector 'a-b' is tan = -10.0 m/-2.0 m = tan-1(5.0) = 78.69o (or) 258.69o __________________________________________________ __________________________________________________ f) The angle between the two vectors is 180°.
______________________________________________ ______________________________________________ c) The sum of the two vectors is given by a+b = [(2.0 m)i^-(2.0 m)j^]+[(4.0 m)i^+(8.0 m)j^] = (6.0 m)i^+(6.0 m)j^ The magnitude of the vector 'a+b' is given by |a+b| = (6.0 m)2+(6.0 m)2 = 8.485 m = 8.49 m The direction of the vector 'a+b' is tan = 6.0 m/6.0 m = tan-1(1.0) = 45o ______________________________________________ ______________________________________________ d) The difference between vector b and vector a is b-a = [(4.0 m)i^+(8.0 m)j^]-[(2.0 m)i^-(2.0 m)j^] = (2.0 m)i^+(10.0 m)j^ The magnitude of the vector 'b-a' is given by |b-a| = (2.0 m)2+(10.0 m)2 = 10.198 m = 10.20 m The direction of the vector 'b-a' is tan = 10.0 m/2.0 m = tan-1(5.0) = 78.69o _________________________________________________ _________________________________________________ e) The difference between vector a and vector b is a-b = [(2.0 m)i^-(2.0 m)j^]-[(4.0 m)i^+(8.0 m)j^] = (-2.0 m)i^-(10.0 m)j^ The magnitude of the vector 'a-b' is given by |a-b| = (-2.0 m)2+(-10.0 m)2 = 10.198 m = 10.20 m The direction of the vector 'a-b' is tan = -10.0 m/-2.0 m = tan-1(5.0) = 78.69o (or) 258.69o __________________________________________________ __________________________________________________ f) The angle between the two vectors is 180°.
______________________________________________ ______________________________________________ d) The difference between vector b and vector a is b-a = [(4.0 m)i^+(8.0 m)j^]-[(2.0 m)i^-(2.0 m)j^] = (2.0 m)i^+(10.0 m)j^ The magnitude of the vector 'b-a' is given by |b-a| = (2.0 m)2+(10.0 m)2 = 10.198 m = 10.20 m The direction of the vector 'b-a' is tan = 10.0 m/2.0 m = tan-1(5.0) = 78.69o _________________________________________________ _________________________________________________ e) The difference between vector a and vector b is a-b = [(2.0 m)i^-(2.0 m)j^]-[(4.0 m)i^+(8.0 m)j^] = (-2.0 m)i^-(10.0 m)j^ The magnitude of the vector 'a-b' is given by |a-b| = (-2.0 m)2+(-10.0 m)2 = 10.198 m = 10.20 m The direction of the vector 'a-b' is tan = -10.0 m/-2.0 m = tan-1(5.0) = 78.69o (or) 258.69o __________________________________________________ __________________________________________________ f) The angle between the two vectors is 180°.
_________________________________________________ _________________________________________________ e) The difference between vector a and vector b is a-b = [(2.0 m)i^-(2.0 m)j^]-[(4.0 m)i^+(8.0 m)j^] = (-2.0 m)i^-(10.0 m)j^ The magnitude of the vector 'a-b' is given by |a-b| = (-2.0 m)2+(-10.0 m)2 = 10.198 m = 10.20 m The direction of the vector 'a-b' is tan = -10.0 m/-2.0 m = tan-1(5.0) = 78.69o (or) 258.69o __________________________________________________ __________________________________________________ f) The angle between the two vectors is 180°.
__________________________________________________ __________________________________________________ f) The angle between the two vectors is 180°.
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