To find
u +
v , we first draw the vector
u , and from the terminal end of
u , we drawn the vector
v . In other words, we have the initial point of
v meet the terminal end of
u . This position corresponds to the notion that we move along the first vector and then, from its terminal point, we move along the second vector. The sum
u +
v is the resultant vector because it results from addition or subtraction of two vectors. The resultant vector travels directly from the beginning of
u to the end of
v in a straight path, as shown in
[link] .
Vector subtraction is similar to vector addition. To find
u −
v , view it as
u + (−
v ). Adding −
v is reversing direction of
v and adding it to the end of
u . The new vector begins at the start of
u and stops at the end point of −
v . See
[link] for a visual that compares vector addition and vector subtraction using
parallelograms .
Adding and subtracting vectors
Given
$u$$=\u27e83,-2\u27e9$ and
$v$$=\u27e8\mathrm{-1},4\u27e9,$ find two new vectors
u +
v , and
u −
v .
To find the sum of two vectors, we add the components. Thus,
To find the difference of two vectors, add the negative components of
$\text{\hspace{0.17em}}v\text{\hspace{0.17em}}$ to
$\text{\hspace{0.17em}}u.\text{\hspace{0.17em}}$ Thus,
While adding and subtracting vectors gives us a new vector with a different magnitude and direction, the process of multiplying a vector by a
scalar , a constant, changes only the magnitude of the vector or the length of the line. Scalar multiplication has no effect on the direction unless the scalar is negative, in which case the direction of the resulting vector is opposite the direction of the original vector.
Scalar multiplication
Scalar multiplication involves the product of a vector and a scalar. Each component of the vector is multiplied by the scalar. Thus, to multiply
$v$$=\u27e8a,b\u27e9$ by
$k$ , we have
$$kv=\u27e8ka,kb\u27e9$$
Only the magnitude changes, unless
$\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is negative, and then the vector reverses direction.
Performing scalar multiplication
Given vector
$\text{\hspace{0.17em}}v$$=\u27e83,1\u27e9,\text{\hspace{0.17em}}$ find 3
v ,
$\frac{1}{2}$$v,\text{\hspace{0.17em}}$ and −
v .
See
[link] for a geometric interpretation. If
$\text{\hspace{0.17em}}v$$=\u27e83,1\u27e9,$ then
In some applications involving vectors, it is helpful for us to be able to break a vector down into its components. Vectors are comprised of two components: the horizontal component is the
$\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ direction, and the vertical component is the
$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ direction. For example, we can see in the graph in
[link] that the position vector
$\u27e82,3\u27e9$ comes from adding the vectors
v_{1} and
v_{2} . We have
v_{1} with initial point
$\text{\hspace{0.17em}}\left(0,0\right)\text{\hspace{0.17em}}$ and terminal point
$\text{\hspace{0.17em}}\left(2,0\right).\text{\hspace{0.17em}}$
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0
then
4x = 2-3
4x = -1
x = -(1÷4) is the answer.
Jacob
4x-2+3
4x=-3+2
4×=-1
4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3
4x=-3+2
4x=-1
4x÷4=-1÷4
x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was 1350 bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after 3 hours?
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?