MAGNITUDE OF A CARTESIAN VECTOR

Hasil carian imej untuk magnitude of a cartesian vector

Example 1

Find the magnitude of the vector = 42+ 3k, the unit vector in the direction of and the vector of magnitude 5 in the direction opposite to x.
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Solution

We have |x=  V~ -2--------2---2-   4  + (- 2) + 3 =  V~ ----------   16 + 4 + 9 =  V~ ---   29.
Therefore the unit vector in the direction of is the vector
  1  V~ ---(4i - 2j + 3k)   29
and the vector of magnitude 5 in the direction opposite to is
 V~ 1-(- 20i + 10j - 15k)   29

Example 2

Given the vectors = 3+ 2and = 2+ 3k, find Cartesian forms of the vectors vv2+ 3v.
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Solution

First,
u + v =  (3i- j + 2k) + (2i + 3j - k)       =  5i + 2j + k.
Next,
u - v =  (3i- j + 2k) - (2i + 3j - k)        =  i- 4j + 3k.
and finally,
2u + 3v =  2(3i- j + 2k) + 3(2i + 3j- k)         =  12i + 7j + k.

Example 3

Given that P1(-123) and P2(334) are two points in space, find the Cartesian form of the vector -P-1-->P2.
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Solution

-- --> P1P2  = (3-  (-1))i + (3- 2)j + (4 - 3)k       = 4i + j + k

Example 4

Show that the vectors
(3 + 4s)i + (4 - 3s)j + 5tk
and
(3s + 4t)i + 5j + (4s - 3t)k
have the same magnitude.
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Solution

Both vectors are in Cartesian form and their lengths can be calculated using the formula
                 V~ ------------ |xi + yj + zk|=    x2 + y2 + z2.
We have
                               V~  ---------------------------- |(3 + 4s)i + (4 - 3s)j + 5tk |=   (3 + 4s)2 + (4 - 3s)2 + (5t)2                                V~ --------------------------------------                             =   9 + 24s + 16s2 + 16 - 24s + 9s2 + 25t2                                V~ ---------2-----2                             =   25 + 25s  + 25t
and
                                V~  --------------------------- |(3s + 4t)i + 5j + (4s - 3t)k |=    (3s + 4t)2 + 52 + (4s - 3t)2                                 V~ --2------2------------------2----2--------                              =  V~  9s-+-16t-+--24st + 25 + 16s +  9t - 24st                              =   25 + 25s2 + 25t2
Therefore two given vectors have the same length.

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