Vector problem solutions (From May 14 post):
(b):B - A = (-i -j) - (3i - 4j) = - 4i + 3 j with magnitude: Ö {(-4) 2 + (3) 2} = Ö 25 and
direction - 38.6 deg with respect to the - x - axis.
2) Using the method of components, find the vector sum of the two vectors A and B if A makes an angle of 45 degrees with the x-axis and has a length of 6 units, and vector B makes an angle of 135 degrees with the x-axis and has a length of 8 units.
Soln. We know: A x = A cos q , A y = A sin q ,
A x = 6 cos 45 o = 6 / Ö2 , A y = 6 sin 45 o = 6 / Ö2
Also: B x = B cos q , B y = B sin q
Whence:
3) For the 3D rectangular box shown earlier, e.g.
For which: T = a i + b j + c k, and assuming sides a = 6, b = 3, and c = 2 :
a)Find the direction cosines.
1) For two vectors A = 3i – 4j and B = -i – j, find the magnitude and direction of:
A + B, and B – A.
A + B, and B – A.
Solns (a).; A + B = (3i - 4j) + (-i -j) = 2i - 5j with magnitude: Ö {(2) 2 + (-5) 2} = Ö 29 and direction - 68.2 deg with respect to +x- axis.
(b):B - A = (-i -j) - (3i - 4j) = - 4i + 3 j with magnitude: Ö {(-4) 2 + (3) 2} = Ö 25 and
direction - 38.6 deg with respect to the - x - axis.
2) Using the method of components, find the vector sum of the two vectors A and B if A makes an angle of 45 degrees with the x-axis and has a length of 6 units, and vector B makes an angle of 135 degrees with the x-axis and has a length of 8 units.
Soln. We know: A x = A cos q , A y = A sin q ,
A x = 6 cos 45 o = 6 / Ö2 , A y = 6 sin 45 o = 6 / Ö2
Also: B x = B cos q , B y = B sin q
Whence:
B x = 8 cos 135 o = - 8 / Ö2 , B y = 8 sin 135 o = 8 / Ö2
3) For the 3D rectangular box shown earlier, e.g.
For which: T = a i + b j + c k, and assuming sides a = 6, b = 3, and c = 2 :
a)Find the direction cosines.
cos a = a / Ö (a 2 + b 2 + c 2 ) = 6 / Ö (6 2 + 3 2 + 2 2 )
= 6 / Ö (36 + 9 + 4 ) = 6 / Ö 49 = 6/7 = 0.857
= 6 / Ö (36 + 9 + 4 ) = 6 / Ö 49 = 6/7 = 0.857
cos b = b / Ö (a 2 + b 2 + c 2 ) = 3 / Ö (6 2 + 3 2 + 2 2 )
= 3 / Ö (36 + 9 + 4 ) = 3 / Ö 49 = 3 /7 = 0.429
cos g = b / Ö (a 2 + b 2 + c 2 ) = 2 / Ö (6 2 + 3 2 + 2 2 )
= 2 / Ö (36 + 9 + 4 ) = 2 / Ö 49 = 2 /7 = 0.286
= 2 / Ö (36 + 9 + 4 ) = 2 / Ö 49 = 2 /7 = 0.286
b) Show that: cos 2 a + cos 2 b + cos 2 g = 1
I.e.
(0.857) 2 + (0.429)2 + ( 0.286)2 = 1
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