Question
Question: If \(\overrightarrow a = \widehat i + \widehat j + \widehat k,\overrightarrow b = \widehat {4i} - \w...
If a=i+j+k,b=4i−2j+3k,c=i−2j+k . Find a vector of magnitude 6 units which is parallel to the vector 2a−b−3c.
Solution
We are given three vectors and let r=2a−b−3cand substituting the given vectors we get the new vector r and in order to find the unit vector we use the formula r=rrand the magnitude formula is given as (coefficient of i)2+(coefficient of j)2+(coefficient of k)2and after obtaining the unit vector the required vector is obtained by multiplying 6 with the unit vector.
Complete step by step solution:
Let r=2a−b−3c
We are given that a=i+j+k,b=4i−2j+3k,c=i−2j+k
Substituting this in we get
⇒r=2(i+j+k)−(4i−2j+3k)−3(i−2j+k) ⇒r=2i+2j+2k−4i+2j−3k−3i+6j−3k ⇒r=−5i+10j−4k
Now the unit vector is given by the formula
⇒r=rr ……….(1)
Where ris the given vector and ris the magnitude of r
The magnitude of a vector is given by (coefficient of i)2+(coefficient of j)2+(coefficient of k)2
Therefore the modulus of is given as
Using this in (1) we get
⇒r=141−5i+10j−4k
Since we are given the magnitude of the required vector is 6
The vector parallel to r with magnitude 6 is given by 6×r
⇒6×141−5i+10j−4k ⇒141−30i+60j−24k
Therefore the required vector is 141−30i+60j−24k.
Note :
Vectors are parallel if they have the same direction. Both components of one vector must be in the same ratio to the corresponding components of the parallel vector.
A unit vector is a vector of length 1, sometimes also called a direction vector.