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 :)
To solve these questions, we need to understand the given information and use vector algebra to calculate the required values. Let's go through each question step by step:
(i) The question asks us to find the point on vector line ON where |ON| is the shortest. Here, O is the origin, and N is a point with an unknown position vector. To find the point, we can minimize the length of vector ON. The formula given for the point is a - (a.b)b.
To find the position vector of N that minimizes |ON|, we can think of the vector a - (a.b)b as the projection of vector a onto the vector b. Basically, we project vector a onto vector b, and by subtracting this projection from vector a, we get the position vector of N.
So, the position vector of N is given by N = a - (a.b)b.
(ii) The question asks us to find the constant k when the position vector of M is a - 5(a.b)b. Here, we need to find the value of k such that kAN = NM.
To find k, we know that kAN is a scalar multiple of AN, meaning both vectors are in the same direction. So, we can equate their components.
Equating the x-components: k(a.x) = (a.x) - 5(a.b)b.x
Equating the y-components: k(a.y) = (a.y) - 5(a.b)b.y
Here, a.x and a.y represent the components of vector a, and b.x and b.y represent the components of vector b.
Using the given information |a| = 2 and a.b = 1/3, substitute these values into the equation above to find k.
(iii) The question asks us to find the geometric meaning of |b x (a-b)| and its exact value. Here, b and (a-b) are both vectors, and the term b x (a-b) represents the cross product of these vectors.
The magnitude of the cross product b x (a-b) represents the area of the parallelogram formed by the vectors b and (a-b). Geometrically, it represents the signed area of the parallelogram. The sign of the area depends on the orientation of the vectors.
To calculate the exact value, we need the components of vectors a and b. Unfortunately, the given information does not provide that. So, we cannot find the exact value without further information.
(iv) The question asks us to write a possible position vector of point C, given that OC bisects the angle AOB. Here, O is the origin, and A and B are vectors.
To find a possible position vector of point C, we need to find the midpoint of vectors A and B. The position vector of the midpoint is the average of their position vectors.
The formula for the position vector of the midpoint M is given by M = (A + B) / 2.
Using this formula, substitute the position vectors of vectors A and B to find a possible position vector of point C.
Note: It's important to note that there may be multiple possible position vectors for point C, as the bisector can be located at different positions within the angle AOB.
I hope this helps! Let me know if you have any further questions.