If α, β, γare angles made by a line with the positive...

If $\alpha, \beta, \gamma$ are angles made by a line with the positive directions of $x, y, z$ axes, then $\left|\begin{array}{ccc}-\sin ^{2} \alpha & \cos \alpha \cos \beta & \cos \alpha \cos \gamma \\ \cos \alpha \cos \beta & -\sin ^{2} \beta & \cos \beta \cos \gamma \\ \cos \alpha \cos \gamma & \cos \beta \cos \gamma & -\sin ^{2} \gamma\end{array}\right|$ (a) 1 (b) $-1$ (c) 0 (d) 3

Key Concepts

Direction Cosines

Direction cosines are the cosines of the angles that a line makes with the positive coordinate axes. They satisfy the relation that the sum of their squares equals one. This concept is pivotal in understanding the orientation of lines in space, as it converts geometric information into algebraic form, facilitating further analysis in problems involving spatial coordinates.

Determinant of a Matrix

The determinant is a scalar value computed from a square matrix that encapsulates key properties of the associated linear transformation, such as invertibility and the scaling of volume in the space. Evaluating the determinant of a matrix with structured entries, especially those derived from geometric considerations, can reveal underlying symmetries and invariances in the system.

Trigonometric Identities

Trigonometric identities, like the Pythagorean identity sin²? + cos²? = 1, are essential in simplifying and transforming expressions involving sine and cosine functions. These identities are crucial when working with direction cosines, aiding in the reduction of complex trigonometric expressions and linking geometric interpretations with algebraic results.

Transcript

00:01 So in the given question we have a matrix in which there are some values and we are told that alpha, beta and gamma are the angles are angles made by a line made by a line made by a line with the positive directions of positive directions of x, y and z axis.

00:43 So this is given in the question.

00:47 So what we have as the determinant that we should evaluate is in the determinant we have minus sine square alpha cos alpha times cos beta cost alpha times cost gamma and in the second row in the second row we have cos alpha cos beta minus sine square beta cos beta cos gamma in the third row we have cos alpha cos gamma cos beta cos gamma and minus sine square gamma so this is the determinant we should evaluate.

01:51 So what we are given is that alpha, beta and gamma are angles made with the positive direction of made by a line with the positive direction of x, y and z, right? so a property of the direction angles made by a line is that we can take the sum of cos square alpha plus cost square beta plus cost square gamma and we will get a value which is equal to 1.

02:25 So this is a property of the direction angles made by a line.

02:30 So let's keep this relation in mind and let's carry on by simplifying the determinant first in order to find the value, right? so first what we are going to do is to write sine square alpha.

02:46 We are going to write this as the sine square terms in this determinant in the terms of cause square right so what we have over here is in the determinant what we can do we can take this and you drag this determinant over here and the change that you are going to make is instead of of sine square we can write sine square theta is equal to 1 minus cos square theta, right? so we are going to use this identity of sine square theta.

03:43 Here we have minus sine square theta.

03:46 So what minus sine square theta would be equal to cos square theta minus 1.

03:53 Right so we are going to make this substitution for the sign square terms in this determinant so we will have cos square cost square alpha minus 1 cost square beta minus 1 and cos square gamma minus 1 right and in the next step what we are going to do is to we can take the cost we can divide the first row with cos alpha we can divide the second row with cos beta and we can divide the third row with cos gamma right so r1 changes to r1 by cos alpha r2 changes to r2 by cos alpha and r3 changes to r3 by cos alpha, cos gamma.

04:54 Cos gamma for r3, cost beta for r2 and cos alpha for r1...

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