The position vector of the point A(3,4) relative to an origin O is a. (a) Express vector a or (OA) in terms of i and j. (b) Find the magnitude of a. (c) Find the unit vector in the direction of a.
(a) 3i+4j
(b) 5
(c) 3/5 i + 4/5 j
(a) The position vector (OA) can be expressed in terms of the unit vectors i and j as follows:
(OA) = 3i + 4j
(b) To find the magnitude of vector a or (OA), we use the formula:
|a| = sqrt((3)^2 + (4)^2)
|a| = sqrt(9 + 16)
|a| = sqrt(25)
|a| = 5
Therefore, the magnitude of vector a or (OA) is 5.
(c) To find the unit vector in the direction of a, we divide vector a or (OA) by its magnitude:
Unit vector in the direction of a = (OA) / |a|
Unit vector in the direction of a = (3i + 4j) / 5
Therefore, the unit vector in the direction of a is (3/5)i + (4/5)j.
(a) To express the position vector a or OA in terms of i and j, we can use the components of the vector. The x-component represents the displacement in the horizontal direction (i direction), and the y-component represents the displacement in the vertical direction (j direction).
Given the point A(3, 4), the x-component is 3 and the y-component is 4. Therefore, the position vector a or OA can be expressed as:
a = 3i + 4j
(b) To find the magnitude of a, we can use the formula for the magnitude of a vector:
Magnitude = √(x^2 + y^2)
In this case, the x-component is 3 and the y-component is 4. Substituting the values into the formula, we have:
Magnitude = √(3^2 + 4^2)
= √(9 + 16)
= √25
= 5
Therefore, the magnitude of vector a is 5.
(c) To find the unit vector in the direction of a, we divide the vector by its magnitude.
Unit vector = (1/Magnitude) * a
Since the magnitude of a is 5, the unit vector in the direction of a can be found as:
Unit vector = (1/5) * (3i + 4j)
= (3/5)i + (4/5)j
Therefore, the unit vector in the direction of a is (3/5)i + (4/5)j.