Finding the Area of a Triangle Using Vectors

The question asks us to find the area of a triangle formed by two given vectors, \(\overrightarrow a\) and \(\overrightarrow b\), which represent two adjacent sides of the triangle.

The given vectors are:

• \(\overrightarrow a = \widehat i + \widehat j - \widehat k\)
• \(\overrightarrow b = 2\widehat i + 2\widehat j + \widehat k\)

The area of a triangle with adjacent sides represented by vectors \(\overrightarrow a\) and \(\overrightarrow b\) is given by the formula:

Area \( = \frac{1}{2} |\overrightarrow a \times \overrightarrow b|\)

where \(|\overrightarrow a \times \overrightarrow b|\) is the magnitude of the cross product of vectors \(\overrightarrow a\) and \(\overrightarrow b\).

First, let's calculate the cross product \(\overrightarrow a \times \overrightarrow b\):

\(\overrightarrow a \times \overrightarrow b = \begin{vmatrix} \widehat i & \widehat j & \widehat k \\ 1 & 1 & -1 \\ 2 & 2 & 1 \end{vmatrix}\)

Expanding the determinant:

\(\overrightarrow a \times \overrightarrow b = \widehat i ((1)(1) - (-1)(2)) - \widehat j ((1)(1) - (-1)(2)) + \widehat k ((1)(2) - (1)(2))\)

\(\overrightarrow a \times \overrightarrow b = \widehat i (1 + 2) - \widehat j (1 + 2) + \widehat k (2 - 2)\)

\(\overrightarrow a \times \overrightarrow b = 3\widehat i - 3\widehat j + 0\widehat k\)

\(\overrightarrow a \times \overrightarrow b = 3\widehat i - 3\widehat j\)

Next, we need to find the magnitude of this resulting vector, \(\overrightarrow a \times \overrightarrow b\).

\(|\overrightarrow a \times \overrightarrow b| = |3\widehat i - 3\widehat j + 0\widehat k|\)

\(|\overrightarrow a \times \overrightarrow b| = \sqrt{(3)^2 + (-3)^2 + (0)^2}\)

\(|\overrightarrow a \times \overrightarrow b| = \sqrt{9 + 9 + 0}\)

\(|\overrightarrow a \times \overrightarrow b| = \sqrt{18}\)

We can simplify \(\sqrt{18}\):

\(\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}\)

So, \(|\overrightarrow a \times \overrightarrow b| = 3\sqrt{2}\).

Finally, calculate the area of the triangle using the formula:

Area \( = \frac{1}{2} |\overrightarrow a \times \overrightarrow b|\)

Area \( = \frac{1}{2} (3\sqrt{2})\)

Area \( = \frac{3\sqrt{2}}{2}\)

Thus, the area of the triangle is \(\frac{3\sqrt{2}}{2}\).