Understanding Vector Angles and Direction Cosines

The problem asks us to find the value of a trigonometric expression involving the angles a given vector makes with the coordinate axes. These angles are related to the direction cosines of the vector.

Let the given vector be \(\vec{a} = 4 \hat{i} - 8 \hat{j} + \hat{k}\). The angles it makes with the positive x, y, and z axes are \(\alpha\), \(\beta\), and \(\gamma\) respectively.

The direction cosines of a vector \(\vec{a} = a_x \hat{i} + a_y \hat{j} + a_z \hat{k}\) are given by:

• \(\cos \alpha = \frac{a_x}{|\vec{a}|}\)
• \(\cos \beta = \frac{a_y}{|\vec{a}|}\)
• \(\cos \gamma = \frac{a_z}{|\vec{a}|}\)

where \(|\vec{a}|\) is the magnitude of the vector.

Calculating Vector Magnitude and Direction Cosines

First, let's calculate the magnitude of the vector \(\vec{a}\):

\(|\vec{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2}\)

For the given vector \(\vec{a} = 4 \hat{i} - 8 \hat{j} + \hat{k}\), we have \(a_x = 4\), \(a_y = -8\), and \(a_z = 1\).

So, the magnitude is:

\(|\vec{a}| = \sqrt{4^2 + (-8)^2 + 1^2} = \sqrt{16 + 64 + 1} = \sqrt{81} = 9\)

Now, we can find the direction cosines:

• \(\cos \alpha = \frac{4}{9}\)
• \(\cos \beta = \frac{-8}{9}\)
• \(\cos \gamma = \frac{1}{9}\)

Using Direction Cosines to find cos2β + cos2γ

We need to find the value of \(\cos 2\beta + \cos 2\gamma\). We can use the double angle identity for cosine: \(\cos 2\theta = 2\cos^2 \theta - 1\).

Applying this identity for \(\beta\) and \(\gamma\):

• \(\cos 2\beta = 2\cos^2 \beta - 1\)
• \(\cos 2\gamma = 2\cos^2 \gamma - 1\)

Substitute the values of \(\cos \beta\) and \(\cos \gamma\):

• \(\cos 2\beta = 2\left(\frac{-8}{9}\right)^2 - 1 = 2\left(\frac{64}{81}\right) - 1 = \frac{128}{81} - 1 = \frac{128 - 81}{81} = \frac{47}{81}\)
• \(\cos 2\gamma = 2\left(\frac{1}{9}\right)^2 - 1 = 2\left(\frac{1}{81}\right) - 1 = \frac{2}{81} - 1 = \frac{2 - 81}{81} = \frac{-79}{81}\)

Now, sum these values:

\(\cos 2\beta + \cos 2\gamma = \frac{47}{81} + \frac{-79}{81} = \frac{47 - 79}{81} = \frac{-32}{81}\)

Alternative Method using Direction Cosine Identity

We know the identity relating the squares of direction cosines: \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\).

There is also a related identity for the double angles: \(\cos 2\alpha + \cos 2\beta + \cos 2\gamma = -1\).

From this identity, we can write: \(\cos 2\beta + \cos 2\gamma = -1 - \cos 2\alpha\).

First, calculate \(\cos 2\alpha\):

\(\cos 2\alpha = 2\cos^2 \alpha - 1 = 2\left(\frac{4}{9}\right)^2 - 1 = 2\left(\frac{16}{81}\right) - 1 = \frac{32}{81} - 1 = \frac{32 - 81}{81} = \frac{-49}{81}\)

Now, substitute this into the identity:

\(\cos 2\beta + \cos 2\gamma = -1 - \left(\frac{-49}{81}\right) = -1 + \frac{49}{81} = \frac{-81 + 49}{81} = \frac{-32}{81}\)

Both methods yield the same result.

The value of \(\cos 2\beta + \cos 2\gamma\) is \(\frac{-32}{81}\).

Revision Table: Vector Direction Cosines

Additional Information on Vector Angles and Trigonometric Identities

Understanding vector direction cosines and related trigonometric identities is crucial in 3D vector algebra. The angles \(\alpha\), \(\beta\), and \(\gamma\) are often called the direction angles of the vector.

The identity \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\) signifies that the direction cosines are not independent; knowing two allows you to find the third (up to sign).

The identity \(\cos 2\alpha + \cos 2\beta + \cos 2\gamma = -1\) is a direct consequence of the \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\) identity and the double angle formula \(\cos 2\theta = 2\cos^2 \theta - 1\). Substituting \(\cos^2 \theta = (\cos 2\theta + 1)/2\) into the first identity gives:

\(\frac{\cos 2\alpha + 1}{2} + \frac{\cos 2\beta + 1}{2} + \frac{\cos 2\gamma + 1}{2} = 1\)

\(\frac{\cos 2\alpha + \cos 2\beta + \cos 2\gamma + 3}{2} = 1\)

\(\cos 2\alpha + \cos 2\beta + \cos 2\gamma + 3 = 2\)

\(\cos 2\alpha + \cos 2\beta + \cos 2\gamma = 2 - 3 = -1\)

This confirms the identity used in the alternative method. These identities are useful for solving problems involving the orientation of vectors in space without needing to explicitly calculate the angles themselves.