Determinants and Triangle Area

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Questions and Answers

What is the determinant of the matrix $A = \begin{bmatrix} 1 & 1 & -2 \ 2 & 1 & -3 \ 5 & 4 & -9 \end{bmatrix}$?

-1
2
1
0 (correct)

If $\begin{vmatrix} 2 & 4 \ 5 & 1 \end{vmatrix} = 2$, what is $x$ if $\begin{vmatrix} 2x & 4 \ 5 & x \end{vmatrix} = 2$?

0
3
1
2 (correct)

What value of $x$ satisfies the equation $\begin{vmatrix} x & 2 \ 18 & x \end{vmatrix} = 0$?

0
-6
4
6 (correct)

Using the determinant formula for the area of a triangle, what is the area formed by the points (0,0), (2,0), and (0,2)?

2 (D) Signup and view all the answers

Given the formula for the area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$, what happens to the area if the points are collinear?

Area is zero (C) Signup and view all the answers

What phenomenon occurs when light bends around obstacles or openings?

Diffraction (C) Signup and view all the answers

Which component of the human eye is responsible for refracting light to help focus images onto the retina?

Cornea (A) Signup and view all the answers

What is the primary purpose of using light ray diagrams in optics?

To predict light behavior in optical systems (D) Signup and view all the answers

How does light travel in different materials compared to its speed in a vacuum?

Slower than in a vacuum (C) Signup and view all the answers

What is the significance of the refractive index in relation to light?

It affects how much light bends when entering a material (C) Signup and view all the answers

What happens to light rays when they encounter a medium with a different optical density?

They diverge or converge depending on the change in optical density. (A) Signup and view all the answers

What is the relationship defined by the law of reflection?

The angle of incidence equals the angle of reflection. (B) Signup and view all the answers

What does Snell's law describe?

The relationship between the angles of incidence and refraction and their respective refractive indices. (A) Signup and view all the answers

What phenomenon occurs when the angle of incidence is greater than the critical angle?

Total internal reflection. (B) Signup and view all the answers

Which statement is true regarding the speed of light in different media?

The speed of light varies based on the medium’s optical density. (C) Signup and view all the answers

Which type of reflection occurs on a smooth surface, resulting in a clear image?

Specular reflection. (C) Signup and view all the answers

What is the primary purpose of lenses in optical instruments?

To focus or diverge light through refraction. (B) Signup and view all the answers

What aspect of light is essential for understanding interference phenomena?

The wave nature of light. (B) Signup and view all the answers

Flashcards

Determinant of a 2x2 matrix

A scalar value calculated from a square matrix. For a 2x2 matrix, it is calculated as (ad) - (bc) where the matrix is [a, b; c, d].

Determinant of a 3x3 matrix

A scalar calculated from a 3x3 matrix using a method that involves expanding along a row or column. It can be tedious, but the concept is based on a series of 2x2 matrix determinants.

Area of a triangle

The area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ can be calculated using the absolute value of one-half of the determinant of the 3x3 matrix [x1,y1,1; x2,y2,1; x3,y3,1]