Vectors Operations Quiz

Explanation
If v = 〈a,b〉, then a is the horizontal component of v. This is because in a two-dimensional Cartesian coordinate system, the x-axis represents the horizontal direction. The vector v = 〈a,b〉 represents a displacement in the x-axis (horizontal direction) of a units and in the y-axis (vertical direction) of b units. Therefore, a represents the displacement in the horizontal direction, making it the horizontal component of v.

Explanation
The given question asks to find the value of 2w + y, where w = and y = . To find 2w, we multiply each component of w by 2, resulting in . Adding this to y, we get + = . Therefore, the correct answer is .

Explanation
The correct answer is "<-5√3, -5>". This is the component form of a vector with a magnitude of 10 and a direction of 150 degrees. The first component, -5√3, represents the horizontal displacement of the vector, and the second component, -5, represents the vertical displacement.

Explanation
To determine if two vectors are orthogonal, their dot product must be equal to zero.
The dot product of u and v is given by -1(3) + 3(y) + 5(z).
Setting this expression equal to zero and solving for y and z, we get 3y + 5z = 1.
The only values of y and z that satisfy this equation are y = 6 and z = -3. Therefore, y = 6 and z = -3 make the vectors u and v orthogonal.

Explanation
To find 2u - 3v, we need to multiply each component of u by 2 and each component of v by -3, and then subtract the corresponding components.

For u = , multiplying each component by 2 gives us .
For v = , multiplying each component by -3 gives us .

Now, subtracting the corresponding components gives us = .

Therefore, the correct answer is .

Explanation
The turtle moves from coordinate P = (3,2) to Q = (5,6). To find the vector that models the turtle's motion, we need to find the difference between the coordinates of Q and P. The x-coordinate of Q is 5 and the x-coordinate of P is 3, so the x-component of the vector is 5 - 3 = 2. Similarly, the y-coordinate of Q is 6 and the y-coordinate of P is 2, so the y-component of the vector is 6 - 2 = 4. Therefore, the vector that models the turtle's motion is .

Explanation
To find 2v + 3w, we need to multiply each component of v by 2 and each component of w by 3, and then add the corresponding components together.

2v = 2 * =
3w = 3 * =

Adding the corresponding components:
2v + 3w = + = =

Therefore, the correct answer is .

Explanation
A vector has both magnitude and direction. Magnitude refers to the length or size of the vector, while direction indicates the orientation or angle at which the vector is pointing. Vectors are often represented by arrows, with the length of the arrow representing the magnitude and the direction of the arrow indicating the direction. Scalars, on the other hand, only have magnitude and do not have a specific direction associated with them. Therefore, the correct answer is vector.

Explanation
The magnitude of a vector is calculated by taking the square root of the sum of the squares of its components. In this case, the vector has a magnitude of √(2^2 + (-3)^2) = √(4 + 9) = √13. Therefore, the correct answer is √13.

Explanation
The angle between two vectors can be found using the dot product formula. The dot product of two vectors A and B is given by A · B = |A| |B| cosθ, where |A| and |B| are the magnitudes of the vectors and θ is the angle between them. In this case, the dot product of and is 3*4 + 4*(-3) = 12 - 12 = 0. Since the dot product is 0, the angle between the two vectors is 90°.