For the vectors 4i +aj+6k and ai +(a-1)j +3k
a.Is there any value of a for which these to vectors will be parallel?
b)Cant these two vectors ever be parallel?
a)
well, the direction must be the same if they are parallel, so
4:a:6 = a : a-1 : 3
4/a = a/(a-1) = 6/3
then 4/a = 6/3 = 2
2a = 4
a = 2
does that work for a/(a-1) = 2 ??
LS = 2/(2-1) = 2
RS = 2
yes, a = 2
b) if they are parallel if a = 2, then they are not parallel for any other value of a
e.g. let a = 5
the first vector is <4, 5, 6> , the 2nd is <5 , 4, 3> , clearly not parallel
a) To determine if two vectors are parallel, we need to check if their direction ratios are proportional.
For the vectors 4i + aj + 6k and ai + (a-1)j + 3k, let's compare the direction ratios of the corresponding components:
First component: 4 and a
Second component: 1 and (a-1)
Third component: 6 and 3
To check if these ratios are proportional, we can set up equations:
4/a = 1/(a-1) = 6/3
Simplifying each ratio, we get:
4/a = 1/(a-1) = 2/1
Cross-multiplying the first equation, we have:
4(a-1) = a
Expanding and simplifying the equation:
4a - 4 = a
Combining like terms:
3a = 4
Dividing both sides by 3:
a = 4/3
So, the two vectors will only be parallel when a = 4/3.
b) The two vectors can be parallel if their direction ratios are proportional. From the previous calculation, we found that the vectors will be parallel if a = 4/3. However, for any other value of a, the vectors will not be parallel.