Gi1a.3d Coordinates System_h? Tr?c T?a ?? Kh�ng Gian

1. What is the Cartesian 3D coordinate system?

Spherical coordinate system

Polar coordinate system

Cylindrical coordinate system

Ox _|_ Oy , Oy _|_ Oz , Oz _|_ Ox

Directional vectors of 3 axes Ox , Oy , Oz are

i , j , k

i = ( 1 , 0 , 0 )
j = ( 0 , 1 , 0 )
k = ( 0 , 0 , 1 )

The Cartesian 3D coordinate system is a system where the axes, Ox, Oy, and Oz are perpendicular to each other. The directional vectors of the axes are represented as i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1). This system is commonly used in mathematics and physics to represent points in a three-dimensional space.

Explanation

The Cartesian 3D coordinate system is a system where the axes, Ox, Oy, and Oz are perpendicular to each other. The directional vectors of the axes are represented as i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1). This system is commonly used in mathematics and physics to represent points in a three-dimensional space.

2. Find the coordinates of point M (2, 3, 2) in vector form.

(3, 2, 2)

Click picture to view the point M OM = 2 i + 3 j + 2 k = 2 ( 1 , 0 , 0 ) + 3 ( 0 , 0 , 1 ) + 2 ( 0 , 0 , 1) = ( 2 , 3 , 2 )

(1, 2, 3)

(0, 3, 2)

The coordinates of a point in vector form are found by breaking down the components of the point into unit vectors i, j, and k and then summing them up accordingly. In this case, the correct coordinates for point M are (2, 3, 2) as shown in the solution.

Explanation

The coordinates of a point in vector form are found by breaking down the components of the point into unit vectors i, j, and k and then summing them up accordingly. In this case, the correct coordinates for point M are (2, 3, 2) as shown in the solution.

3. Determine vector AB from points A and B A ( -2 , 1 , 0 ) B ( 1 , 0 , -1 ). Calculate the modulus of AB.

AB = (xB - xA , yB - yA , zB - zA) AB = (1 - 2 , 1 - 0 , 0 - (-1)) = (-1 , 1 , 1) AB = sqrt(xAB^2 + yAB^2 + zAB^2) AB = sqrt((-1)^2 + 1^2 + 1^2) = sqrt(3).

AB = ( xB - xA , yB - yA , zB - zA )
AB = ( 1 + 2 , 0 - 1 , -1 - 0 ) = ( 3 , -1 , -1 )

AB = sqrt ( xAB2 + yAB2 + zAB2 )
AB = sqrt( 32 + (-1)2 + (-1)2 ) = sqrt ( 11 )

AB = (xB - xA , yB - yA , zB - zA) AB = (1 - 2 , 0 - 1 , -1 - 0) = (-1 , -1 , -1) AB = sqrt(xAB^2 + yAB^2 + zAB^2) AB = sqrt((-1)^2 + (-1)^2 + (-1)^2) = sqrt(3).

AB = (xB - xA , yB - yA , zB - zA) AB = (1 - (-2) , 0 - 1 , -1 - 0) = (3 , -1 , -1) AB = sqrt(xAB^2 + yAB^2 + zAB^2) AB = sqrt(3^2 + (-1)^2 + (-1)^2) = sqrt(11).

The correct formula to determine the vector AB is subtracting the corresponding coordinates of points B and A. The modulus of AB is calculated by taking the square root of the sum of squares of the vector components.

Explanation

The correct formula to determine the vector AB is subtracting the corresponding coordinates of points B and A. The modulus of AB is calculated by taking the square root of the sum of squares of the vector components.

4. Given a = ( 2 , -1 , 3 ) ; b = ( 3 , -1 , -1 ) Calculate a + b (sum) ; a - b (subtract) ; a . b (dot product - scalar product) ; a x b (cross product-vector product).

A) a - b = ( 2 , -1 , 3 ) - ( 3 , -1 , -1 ) = [-1, 0, 4]

K?t qu? c?a a + b ( T?ng 2 vector ) a + b = ( 2 , -1 , 3 ) + ( 3 , -1 , -1 ) = [5, -2, 2]
Click picture to view result .
2 vect? c?ng nhau : (h+h,t+t,c+c)

C) a x b = (2*-1 - 3*-1), (3-1 - 3*2), (-1*1 - 2*3) = (-1, -5, -7)

B) a . b = ( 2 , -1 , 3 ) . ( 3 , -1 , -1 ) = 8

The correct answer for addition of vectors is to sum the corresponding components of the vectors. The subtraction involves subtracting corresponding components. The dot product is calculated by multiplying corresponding components and summing them up. The cross product results in a new vector which is perpendicular to the original two vectors.

Explanation

The correct answer for addition of vectors is to sum the corresponding components of the vectors. The subtraction involves subtracting corresponding components. The dot product is calculated by multiplying corresponding components and summing them up. The cross product results in a new vector which is perpendicular to the original two vectors.

5. What is the result of a - b?

Kết quả của a - b (Tích vô hướng)

K?t qu? c?a a - b ( Hi?u 2 vector )
a - b = ( 2 , -1 , 3 ) - ( 3 , -1 , -1 ) = [-1, 0, 4]
Click picture to view result .

2 vect? tr? nhau :(h-h,t-t,c-c)

Kết quả của a - b (Phép nhân ma trận)

Kết quả của a - b (Tổng 2 vector)

The correct answer involves finding the difference between the two vectors a and b by subtracting corresponding components.

Explanation

The correct answer involves finding the difference between the two vectors a and b by subtracting corresponding components.

6. GOODBYE

SEE YOU THEN

TAKE CARE

BYE-BYE

ADIÓS

The correct response 'SEE YOU THEN' implies a plan to meet or talk again in the future, while the other incorrect options may also express farewell but do not specifically indicate a future interaction.

Explanation

The correct response 'SEE YOU THEN' implies a plan to meet or talk again in the future, while the other incorrect options may also express farewell but do not specifically indicate a future interaction.