Final answer:
The vectors a, b, and c form the sides of a right-angled triangle, with c being the hypotenuse. Their lengths satisfy the Pythagorean theorem, thereby proving the triangle is right-angled.
Step-by-step explanation:
To show that the vectors a = 3i, b = 4j, and c = 3i + 4j form a right-angled triangle, we can use the concept that the components of a vector along the x- and y-axes, denoted as Ax and Ay, form a right triangle with vector A.
Vector a has components 3 along the x-axis and 0 along the y-axis, while vector b has components 0 along the x-axis and 4 along the y-axis. When you combine these two vectors, vector c (= a + b) forms the hypotenuse of the right triangle, with its components being exactly the sum of the corresponding components of vectors a and b, which are 3 along the x-axis and 4 along the y-axis, respectively.
The lengths of a, b, and c are |a| = 3, |b| = 4, and |c| = √(3² + 4²) = 5. According to the Pythagorean theorem, since 3² + 4² = 5², these vectors indeed form a right-angled triangle, with vector c being the hypotenuse.