The position vectors for points P and Q are 4 I + 3 j + 6 j + 6 k respectively. Express vector PQ in terms of unit vectors I, j and k. Hence find the length of PQ, in simplified surd form.
It doe not look too clear what P and Q are, I am assuming the following.
P=(4,3,0)
Q=(0,6,6)
PQ = Q-P = <4,-3,-6> by subtraction
= 4i-3j-6k in unit vectors.
Length of vector
<x,y,z> = sqrt(x^2+y^2+z^2)
To express vector PQ in terms of unit vectors i, j, and k, we need to find the difference between the position vectors of points P and Q.
Given that the position vector of point P is 4i + 3j + 6j + 6k, we can combine like terms to simplify it. The j component appears twice, so it becomes 3j + 6j, which simplifies to 9j. Therefore, the position vector of point P is 4i + 9j + 6k.
Similarly, the position vector of point Q is already given as 4i + 6j + 6k.
Next, to find vector PQ, we subtract the position vector of point Q from the position vector of point P:
PQ = (4i + 9j + 6k) - (4i + 6j + 6k)
Simplifying, we can cancel out like terms:
PQ = 4i + 9j + 6k - 4i - 6j - 6k
This leaves us with:
PQ = (4i - 4i) + (9j - 6j) + (6k - 6k)
Finally, simplifying further, we have:
PQ = 0i + 3j + 0k
Therefore, vector PQ is simply 3j.
To find the length of PQ, we use the magnitude formula:
|PQ| = √(y^2)
In this case, y represents the coefficient of j in vector PQ, which is 3:
|PQ| = √(3^2)
Simplifying, we have:
|PQ| = √9
Therefore, the length of PQ is 3 in simplified surd form.