3-D geometry includes many concepts, including the direction cosines of a line, direct ratios of line, straight line, condition of perpendicularity, etc. Generally, the direction cosines of a line consist of making a line with the X and Y axis along with alpha, beta, and gamma.
Consider a line OP going through the beginning.
The points the line OP makes with the x,y, and z makes angles, i.e., α,β and γ. Then, cosα,cosβ, and cosγ are the heading cosines of the line OP.
For lines not going through the beginning, the heading cosines are tracked down utilising the course proportions.
Think about line AB. Presently define the boundary corresponding to line AB going through the beginning, i.e., OP.
Two similar lines have similar heading cosines.
A straight line is a bend, to such an extent that everyone focuses on the line section joining any two places on it.
Three Dimensional Geometry Sample Questions for Class 12
Question 1
Find the angle between the two vectors. vecu=-8vecjvecv=-9veci-2vecj
A. 77.47°
B. 86.42°
C. 74.87°
D. 68.92°
Question 2
Matrices can be used to send coded messages. One way this is done is by taking a message, changing the letters into numbers, and then putting these numbers into a matrix. Zero represents a blank space. The matrix is formed by starting with the upper left most element, and going down each column till its full, then starting on the next column. If there are extra spaces in a matrix, fill the remaining elements with zeros (for example, if a message only has 7 letters, and one uses a 3-by-3 matrix, the remaining 2 elements would be zeros). Call the matrix containing the message matrix M. Then, one multiplies this by an encoding matrix; call this matrix A. Thus, the message sent would be the matrix AM. The recipient knows the encoding matrix, A. All they have to do is to multiply by the inverse, A^{-1}. For the coded message [[24,20,56,76,55,24,20,56,37],[87,78,206,267,185,87,78,206,127],[27,95,173,118,-110,27,95,173,1]], and the coding matrix [[-1,3,4],[-3,10,14],[2,-6,7]], what is the message?
A. KEEP THIS TO YOURSELF ONLY
B. KEEP IT SECRET KEEP IT SAFE
C. DO NOT TELL ANYONE ANYTHING
D. KEEP CONCEALED AT ALL COSTS
Question 3
What is the equation for a circle with center (2,-2) and radius of 2?
A. (x-2)^2+(y+2)^2=4
B. (x-2)^2+(y+2)^2=2
C. (x+2)^2+(y-2)^2=2
D. (x+2)^2+(y-2)^2=4
Question 4
An equation of the line tangent to the graph of y=cos(2x) at x=pi/4 is which of the following?
A. y-1=-(x-pi/4)
B. y-1=-2(x-pi/4)
C. y=2(x-pi/4)
D. y=-(x-pi/4)
Question 5
Given the triangle with vertices (1,1), (2,3), and (5,1), which of the following matrix expressions would represent the reflection of this triangle over the y-axis?
A. [[1,0],[0,-1]] \ [[1,1],[2,3],[5,1]]
B. [[-1,0],[0,1]] \ [[1,2,5],[1,3,1]]
C. [[-1,0],[0,1]] \ [[1,1],[2,3],[5,1]]
D. [[1,0],[0,-1]] \ [[1,2,5],[1,3,1]]