Math: Vectors and Vector Operations
Adding, subtracting, scaling, and interpreting vectors
Math: Vectors and Vector Operations
Adding, subtracting, scaling, and interpreting vectors
Math - Grade 9-12
- 1
Vector a = <3, 4>. Find the magnitude of vector a.
Use the magnitude formula for a two-dimensional vector.
The magnitude of vector a is 5 because sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5. - 2
Let u = <2, -1> and v = <5, 3>. Find u + v.
The sum is <7, 2> because you add corresponding components: 2 + 5 = 7 and -1 + 3 = 2. - 3
Let u = <6, 2> and v = <1, 7>. Find u - v.
Subtract component by component.
The difference is <5, -5> because you subtract corresponding components: 6 - 1 = 5 and 2 - 7 = -5. - 4
Find 3< -2, 5 >.
The result is <-6, 15> because each component is multiplied by 3. - 5
A vector has initial point (1, 2) and terminal point (6, 8). Write the vector in component form.
Subtract the coordinates of the initial point from the terminal point.
The vector is <5, 6> because 6 - 1 = 5 and 8 - 2 = 6. - 6
A plane travels 120 km east and then 50 km north. Represent the trip as a vector in component form.
The vector is <120, 50> if east is positive x and north is positive y. - 7
Find the magnitude of v = <8, 15>.
Square each component, add, and then take the square root.
The magnitude is 17 because sqrt(8^2 + 15^2) = sqrt(64 + 225) = sqrt(289) = 17. - 8
Let p = <-4, 9> and q = <3, -2>. Find 2p + q.
First, 2p = <-8, 18>. Then 2p + q = <-8, 18> + <3, -2> = <-5, 16>. - 9
Determine whether the vectors <4, 6> and <2, 3> are scalar multiples of each other. Explain briefly.
Compare the ratio of corresponding components.
Yes, they are scalar multiples because <4, 6> = 2<2, 3>. Both components are multiplied by the same number, 2. - 10
Find the midpoint of the segment with endpoints A(2, -3) and B(10, 5).
The midpoint is (6, 1) because ((2 + 10) / 2, (-3 + 5) / 2) = (6, 1). - 11
A hiker walks 7 miles west and 24 miles south. Write the displacement vector in component form and find its magnitude.
West and south are negative directions in this coordinate system.
The displacement vector is <-7, -24> if east is positive x and north is positive y. Its magnitude is 25 miles because sqrt((-7)^2 + (-24)^2) = sqrt(49 + 576) = sqrt(625) = 25. - 12
Let a = <1, 2> and b = <-3, 4>. Find a + 2b.
First, 2b = <-6, 8>. Then a + 2b = <1, 2> + <-6, 8> = <-5, 10>. - 13
A vector has magnitude 10 and points directly along the positive x-axis. Write the vector in component form.
A vector on the x-axis has a y-component of 0.
The vector is <10, 0> because all of its length is in the positive x-direction and none is in the y-direction. - 14
Find the distance between the points ( -1, 4 ) and ( 5, -4 ).
The distance is 10 because the change in x is 6 and the change in y is -8, so the distance is sqrt(6^2 + (-8)^2) = sqrt(36 + 64) = sqrt(100) = 10. - 15
Vector u = <x, 7> and vector v = <3, 2>. If u + v = <11, 9>, find x.
Match the components of the sum.
The value of x is 8 because <x, 7> + <3, 2> = <x + 3, 9>. Since x + 3 = 11, x = 8.