Relative to the origin O, two points A and B have position vectors a and b respectively. A line, l, passes through A and is parallel to b. It is given that b is a unit vector.
(i) Write down a vector equation of l. Show that the position vector of the point N on l such that the length |ON| is the shortest is given by a-(a.b)b.
(ii) The point M is on AN produced such that kAN = NM, where k is a constant. Given that the position of M is a-5(a.b)b, find k.
It is given that |a|=2 and a.b=1/3.
(iii) Give the geometric meaning of |b x (a-b)| and find its exact value.
(iv) C is a point such that OC bisects the angle AOB. Write down, in terms of a and b, a possible position vector of C.
Any help would be greatly appreciated! Thank you :)
(i) To write down a vector equation of the line l passing through A and parallel to b, we can use the point-normal form of the equation of a line. The normal vector of the line is b since the line is parallel to b. Let's call the position vector of the point N as n.
The vector equation of the line l passing through A and parallel to b is:
n = a + tb, where t is a scalar parameter.
To show that the position vector of the point N on l such that |ON| is the shortest is given by a - (a.b)b, we need to minimize the length |ON|. For the shortest distance, the vector ON should be perpendicular to the direction vector b.
Since b is a unit vector, we can use the dot product to find a vector perpendicular to b. The dot product of two vectors is zero if and only if the vectors are perpendicular. So, we have:
(a - (a.b)b) . b = 0
Expanding this, we get:
a.b - (a.b)(b.b) = 0
a.b - (a.b) = 0
Simplifying further, we have:
a.b = a.b
Therefore, the position vector a - (a.b)b satisfies the condition for the shortest |ON|.
(ii) Given that the position of point M is a - 5(a.b)b, we can find k by using the fact that kAN = NM. Let's call the position vector of M as m.
The position vector of M is:
m = a - 5(a.b)b
The position vector of N is:
n = a - (a.b)b (as derived in part (i))
Using the fact that kAN = NM, we can write:
n - a = -k(a - n - a)
Substituting the values of n and a, we have:
(a - (a.b)b) - a = -k(a - (a - (a.b)b) - a)
Simplifying this, we get:
(a.b)b = -k(a.b)b
Since b is a unit vector, it is nonzero, so we can divide both sides by (a.b)b:
a.b = -k(a.b)
Given that a.b = 1/3, we can solve for k:
1/3 = -k(1/3)
k = -1
Therefore, the value of k is -1.
(iii) The expression |b x (a - b)| represents the magnitude of the cross product between vectors b and (a - b). Geometrically, the magnitude of the cross product represents the area of the parallelogram formed by the two vectors.
In this case, |b x (a - b)| represents the area of the parallelogram formed by vectors b and (a - b). The exact value can be found by taking the magnitude of the cross product.
(iv) To find a possible position vector of point C such that OC bisects the angle AOB, we need to find the bisector of the angle.
The bisector of the angle AOB is the line passing through the origin O and C and is perpendicular to the line l.
Since the line l is parallel to b, the direction vector of the line perpendicular to l will be perpendicular to b. This can be found by taking the cross product of b with any vector perpendicular to it.
Let's call the vector perpendicular to b as v. We can find v by choosing any vector that is not parallel to b. Let's choose v = a - b.
The position vector of C can be written as:
c = λ(a - b) + δ(b x (a - b))
where λ and δ are scalar parameters.
This equation represents all possible position vectors of C that satisfy OC bisects the angle AOB.