Vector Examples -
Ex. 1 Find the component of the vector, F1 = 3i + 4j in the direction of the vector, F2 = 5i + 12j where i and j are unit vectors in the x and y directions. Solution: The magnitude of the vector F2 is 13 (square root of the sum of the squares of each component). So the unit vector in the direction of F2 is (5/13)i + (12/13)j . Recall from vector calculus the component of a vector, A, in the direction of vector, B, is the dot product, A ● B / | B | F1 ● F2/13 = ( 3i + 4j ) ● [(5/13)i + (12/13)j] = (15/13) + (48/13) = 63/13 So the magnitude of the resulting vector is 63/13. |
Ex. 2 Find the vector of F1 = 3i + 4j in the direction of the vector, F2 where F2 = 5i + 12j and where i and j are unit vectors in the x and y directions. Solution: From the first example the magnitude of the resulting vector in the direction of vector, F2 is 63/13 and the unit vector in the direction of F2 is (5/13)i + (12/13)j . The resulting vector, F3, in the direction of F2 is just its magnitude times the unit vector in the direction of F2 . So F3 = (63/13) [(5/13)i + (12/13)j] = (315/13)i + 756/13)j
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Note the same strategy applies to vectors in three dimensions with cartesian unit vectors i, j, and k. |
Find
the sum of vectors A =
3i + 4j
+ 5k and B = 12i - 16j
+ 22k Strategy: Add components. A + B = (3i
+ 4j + 5k) +
(12i - 16j
+ 22k) = 15i - 12j
+ 27k |
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