Animated Solution for Mathematics - Vector Algebra: Consider points A, B, C and D with position vectors $7\hat{i}-4\hat{j}+7\hat{k},\hat{i}-6\hat{j}+10\hat{k},-\hat{i}-3\hat{j}+4\hat{k}$ and $5\hat{i}-\hat{j}+5\hat{k}$ respectively. Then ABCD is a

Welcome, future engineer. Today we are embarking on a journey through 3D space to uncover the identity of a mysterious quadrilateral. We are given four points, $A,B,C,$ and $D$, defined by their position vectors.

By extracting the coefficients of $\hat{i},\hat{j},$ and $\hat{k}$, we translate these vectors into coordinates: $A=(7,-4,7)$ $B=(1,-6,10)$ $C=(-1,-3,4)$ $D=(5,-1,5)$

The Distance Formula

To identify the quadrilateral, we need to know the lengths of its sides. In 3D space, our most reliable tool is the distance formula:

This formula acts as the 'ruler' of the 3D world, allowing us to measure the distance between any two points. We will use this formula systematically to find the lengths of all four sides: $AB,BC,CD,$ and $DA$.

Let us start with side $AB$. Substituting the coordinates of $A$ and $B$:

Next, we move to side $BC$. Using $B(1,-6,10)$ and $C(-1,-3,4)$:

Now, let us calculate side $CD$. Using $C(-1,-3,4)$ and $D(5,-1,5)$:

Finally, we calculate side $DA$. Using $D(5,-1,5)$ and $A(7,-4,7)$:

We have determined the side lengths: $AB=7$, $BC=7$, $CD=\sqrt{41}$, and $DA=\sqrt{17}$.

For a quadrilateral to be a parallelogram, opposite sides must be equal ($AB=CD$ and $BC=DA$). Here, $7 eq \sqrt{41}$ and $7 eq \sqrt{17}$.

Since the opposite sides are not equal, this quadrilateral is not a parallelogram. Because a rhombus, square, and rectangle are all special types of parallelograms, this shape cannot be any of those either.

The lesson here is clear: never assume the shape's identity. Always let the math guide you to the truth.