(1).answer that correctly shows following vectors,given in component form, in standard position.
(1). V=(5,-2)
(2). v=(-3,-4)
(3).u=(16,-8)
v=(-5,3)
u+v= ? (#,#)
(4).u=(10,5)
-3u= ?
(3) u+v = ((16-5),(-8+3)) = (11,-5)
(4) -3u = (-30,-15)
(3). u+v = (16,-8) + (-5,3) = (16 + (-5), -8 + 3) = (11, -5)
(4). -3u = -3(10,5) = (-30, -15)
To find the vector sum u+v, we add the corresponding components of the vectors u and v.
Given:
u = (16, -8)
v = (-5, 3)
Adding the components:
u + v = (16 + (-5), -8 + 3)
= (11, -5)
Therefore, u + v = (11, -5).
Next,
To find the vector -3u, we multiply each component of vector u by -3.
Given:
u = (10, 5)
Multiplying the components:
-3u = (-3 * 10, -3 * 5)
= (-30, -15)
Therefore, -3u = (-30, -15).
To answer these questions and understand how to get the answer, you need to know some basic concepts related to vectors.
1. Standard Position:
In the standard position, the initial point of the vector is at the origin (0,0). So, to represent vectors in the standard position, we need to move the vector so that its tail is at the origin.
2. Component Form:
Vectors in component form are represented by their x and y-components. For example, (5,-2) represents a vector with an x-component of 5 and a y-component of -2.
Now let's solve the given questions step by step:
1. Given vectors:
(a) v = (5, -2)
To represent this vector in the standard position, start at the origin, and move 5 units to the right (positive x-direction) and 2 units down (negative y-direction).
(b) v = (-3, -4)
In the standard position, start at the origin, and move 3 units to the left (negative x-direction) and 4 units down (negative y-direction).
(c) u = (16, -8)
To represent this vector in the standard position, start at the origin, and move 16 units to the right (positive x-direction) and 8 units down (negative y-direction).
(d) v = (-5, 3)
In the standard position, start at the origin, and move 5 units to the left (negative x-direction) and 3 units up (positive y-direction).
2. u + v = ?
To find the sum of two vectors, add their corresponding components.
For u = (16, -8) and v = (-5, 3), add the x-components and y-components separately:
u + v = (16 + (-5), -8 + 3) = (11, -5)
So, the sum of u and v in standard position is (11, -5).
3. -3u = ?
To scalar multiply a vector, multiply each component of the vector by the scalar.
For u = (10, 5) and scalar -3:
-3u = (-3*10, -3*5) = (-30, -15)
So, -3u in standard position is (-30, -15).
Note: The (#,#) in the questions simply represents the format of the answer, with a pair of values separated by a comma.