Understanding Vector Projection

The projection of a vector \(\overrightarrow b\) onto a vector \(\overrightarrow a\) is a scalar value representing the length of the component of \(\overrightarrow b\) that lies in the direction of \(\overrightarrow a\). It is calculated using the formula:

Projection of \(\overrightarrow b\) on \(\overrightarrow a\) = \(\frac{\overrightarrow a \cdot \overrightarrow b}{||\overrightarrow a||}\)

Here, \(\overrightarrow a \cdot \overrightarrow b\) is the dot product of the two vectors, and \(||\overrightarrow a||\) is the magnitude of vector \(\overrightarrow a\).

Step-by-Step Calculation

We are given the vectors:

• \(\overrightarrow a = 2\widehat i - 2\widehat j - \widehat k\)
• \(\overrightarrow b = 3\widehat i + 2\widehat j + \widehat k\)

Calculate the Dot Product (\(\overrightarrow a \cdot \overrightarrow b\))

The dot product of two vectors \(\overrightarrow a = a_1\widehat i + a_2\widehat j + a_3\widehat k\) and \(\overrightarrow b = b_1\widehat i + b_2\widehat j + b_3\widehat k\) is given by \(a_1b_1 + a_2b_2 + a_3b_3\).

For the given vectors:

\(\overrightarrow a \cdot \overrightarrow b = (2)(3) + (-2)(2) + (-1)(1)\)

\(\overrightarrow a \cdot \overrightarrow b = 6 - 4 - 1\)

\(\overrightarrow a \cdot \overrightarrow b = 1\)

Calculate the Magnitude of Vector \(\overrightarrow a\) (\(|| \overrightarrow a ||\))

The magnitude of a vector \(\overrightarrow a = a_1\widehat i + a_2\widehat j + a_3\widehat k\) is given by \(\sqrt{a_1^2 + a_2^2 + a_3^2}\).

For vector \(\overrightarrow a = 2\widehat i - 2\widehat j - \widehat k\):

\(||\overrightarrow a|| = \sqrt{(2)^2 + (-2)^2 + (-1)^2}\)

\(||\overrightarrow a|| = \sqrt{4 + 4 + 1}\)

\(||\overrightarrow a|| = \sqrt{9}\)

\(||\overrightarrow a|| = 3\)

Calculate the Projection

Now, substitute the calculated values into the projection formula:

Projection of \(\overrightarrow b\) on \(\overrightarrow a\) = \(\frac{\overrightarrow a \cdot \overrightarrow b}{||\overrightarrow a||}\)

Projection = \(\frac{1}{3}\)

Conclusion

The projection of vector \(\overrightarrow b\) on vector \(\overrightarrow a\) is \(\frac{1}{3}\).