A line has slope-3 and y-intercept 5. Find a vector equation of the line.
a. [x, y] = [0, 5] + [1.-3]
b. IX v1 = [5,0] + 110.-31
c. LX. y = [1, -3] + [0, ]
d. x. v = (-3,5] + 1-3, -31
The Correct Answer and Explanation is :
To find the vector equation of a line with a given slope and y-intercept, let’s first break down the problem and understand the components involved.
Step-by-Step Solution:
We are given the following information:
- Slope of the line (m) = -3
- Y-intercept = 5
The equation of a line in slope-intercept form is given as:
[
y = mx + b
]
where:
- ( m ) is the slope,
- ( b ) is the y-intercept.
Substituting the given values:
[
y = -3x + 5
]
This is the equation of the line in terms of (x) and (y).
Now, let’s express this line in vector form. A vector equation of a line can be written as:
[
[x, y] = [x_0, y_0] + t \cdot [a, b]
]
where:
- ([x_0, y_0]) is a point on the line (often the y-intercept or any known point),
- ([a, b]) is a direction vector representing the slope (the change in (x) and (y)),
- (t) is a scalar parameter that scales the direction vector.
Step 1: Identify a point on the line
We know that the y-intercept is 5, so one point on the line is:
[
[x_0, y_0] = [0, 5]
]
This is the point where the line crosses the y-axis.
Step 2: Direction vector based on the slope
The slope of the line is -3. This means for every 1 unit increase in (x), (y) decreases by 3 units. Therefore, the direction vector, ([a, b]), corresponding to the slope is:
[
[a, b] = [1, -3]
]
Here, for each unit increase in (x), (y) changes by -3.
Step 3: Form the vector equation
Now that we have both the point ([0, 5]) and the direction vector ([1, -3]), we can write the vector equation of the line as:
[
[x, y] = [0, 5] + t \cdot [1, -3]
]
This is the vector equation of the line, where (t) is a scalar that can take any real value.
Correct Answer:
From the options provided, the correct vector equation is:
a. ([x, y] = [0, 5] + t \cdot [1, -3])
This is the equation of the line in vector form.
Explanation:
- The point ([0, 5]) is a known point on the line, which is the y-intercept.
- The direction vector ([1, -3]) is derived from the slope of -3, indicating that for each step in the positive (x)-direction, the value of (y) decreases by 3.
Thus, this vector equation correctly describes the line with slope -3 and y-intercept 5.
I will now generate a visual representation of this line for you.
Here is the visual representation of the line with slope -3 and y-intercept 5. The line passes through the point (0, 5) on the y-axis, and its slope causes it to slope downward as it moves along the x-axis. This image shows how the line behaves based on the vector equation we’ve derived.