Question

The points in the Argand plane represented by the complex numbers \(4\hat{i}+3\hat{j}\), \(6\hat{i}-2\hat{j}-3\hat{k}\) and \(\hat{i}-\hat{j}-3\hat{k}\) form

• A. a right - angled triangle
• B. a right - angled isosceles triangle
• C. an equilateral triangle
• D. an isosceles triangle

Hint:

For points represented by complex numbers \(z_A, z_B, z_C\), calculate side lengths \(|z_A-z_B|\), \(|z_B-z_C|\), \(|z_C-z_A|\). Ignoring \(\hat{k}\) components for Argand plane: \(A(4+3i), B(6-2i), C(1-i)\). \(AB^2=29, BC^2=26, CA^2=25\). This is scalene. The question's provided data does not form an isosceles triangle under standard interpretation.

Answer 1

The Correct Option is D

The term "Argand plane" refers to the 2D complex plane.
The notation \(a\hat{i}+b\hat{j}+c\hat{k}\) is for 3D vectors.
This indicates an inconsistency in the question statement.
We will assume the question intends for the points in the Argand plane to be derived from the \(\hat{i}\) (real part) and \(\hat{j}\) (imaginary part) components, and the \(\hat{k}\) components are to be disregarded for an Argand plane representation.
Let the complex numbers be \(z_1, z_2, z_3\): \(z_1\) corresponds to \(4\hat{i}+3\hat{j} \implies z_1 = 4+3i\).
Let this be point A.
\(z_2\) corresponds to \(6\hat{i}-2\hat{j}-3\hat{k} \implies z_2 = 6-2i\).
Let this be point B.
\(z_3\) corresponds to \(\hat{i}-\hat{j}-3\hat{k} \implies z_3 = 1-i\).
Let this be point C.
Calculate the square of the side lengths of the triangle ABC: \(AB^2 = |z_1-z_2|^2 = |(4+3i) - (6-2i)|^2 = |(4-6) + (3-(-2))i|^2 = |-2+5i|^2 \) \(AB^2 = (-2)^2 + (5)^2 = 4+25 = 29\).
\(BC^2 = |z_2-z_3|^2 = |(6-2i) - (1-i)|^2 = |(6-1) + (-2-(-1))i|^2 = |5-i|^2 \) \(BC^2 = (5)^2 + (-1)^2 = 25+1 = 26\).
\(CA^2 = |z_3-z_1|^2 = |(1-i) - (4+3i)|^2 = |(1-4) + (-1-3)i|^2 = |-3-4i|^2 \) \(CA^2 = (-3)^2 + (-4)^2 = 9+16 = 25\).
The side lengths are \(AB=\sqrt{29}\), \(BC=\sqrt{26}\), \(CA=\sqrt{25}=5\).
Since all three side lengths are different, the triangle formed is a scalene triangle.
This result contradicts the provided "Correct Answer" (4) which states it is an isosceles triangle.
There appears to be an error in the question's data or the provided correct option, as standard interpretation does not yield an isosceles triangle.
For the solution to be "an isosceles triangle", two side lengths must be equal.
If we proceed assuming the "Correct Answer" is (4), this implies that the problem intended different numerical values for the points.
Due to this discrepancy, a step-by-step derivation to the provided answer is not possible with the given numbers.
\[ \boxed{\text{an isosceles triangle (Note: Data leads to scalene triangle)}} \]