If vectors v⃗=⟨4,1⟩v ⃗=⟨4,1⟩, u⃗=⟨−8,3⟩u ⃗=⟨-8,3⟩u⃗=⟨−8,3⟩, and w⃗=⟨−2,−1⟩w ⃗=⟨-2,-1⟩w⃗=⟨−2,−1⟩, calculate w⃗−3(v⃗+u⃗)w ⃗-3(v ⃗+u ⃗)w⃗−3(v⃗+u⃗).

⟨10,−13⟩\langle10,-13\rangle⟨10,−13⟩

⟨10,13⟩\langle10,13\rangle⟨10,13⟩

⟨−14,13⟩\langle-14,13\rangle⟨−14,13⟩

⟨−14,−13⟩\langle-14,-13\rangle⟨−14,−13⟩

9. Vectors

Vectors in Component Form

9. Vectors

Vectors in Component Form : Videos & Practice Problems

Video Lessons Worksheet

Concept

Position Vectors & Component Form

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Position Vectors & Component Form Video Summary

Understanding vectors is crucial in mathematics, particularly in geometry and physics. Vectors can be visualized as arrows in space, which can be manipulated by stretching, shrinking, or combining them to form resultant vectors. A key concept in vector analysis is the position vector, which is defined as a vector that originates from the coordinate system's origin. This means that to represent a vector as a position vector, you simply adjust its starting point to the origin of the graph.

To express vectors numerically, we use component form, which breaks down the vector into its x and y components. For instance, if a vector moves 3 units to the right and 2 units up, it can be represented as $\mathbf{v} = (3, 2)$. The first number indicates the movement along the x-axis, while the second number indicates the movement along the y-axis. This method of representation is straightforward and intuitive.

In cases where the initial and terminal points of a vector are given, you can still determine the vector's component form without graphing it. The formula to find the vector $\mathbf{v}$ is:

$\mathbf{v} = (x_f - x_i, y_f - y_i)$

Here, $x_f$ and $y_f$ are the coordinates of the terminal point, while $x_i$ and $y_i$ are the coordinates of the initial point. For example, if the initial point is (0.2, 0.3) and the terminal point is (0.3, 0.5), the vector can be calculated as follows:

$\mathbf{v} = (0.3 - 0.2, 0.5 - 0.3) = (0.1, 0.2)$

This result indicates that the vector moves 0.1 units in the x-direction and 0.2 units in the y-direction. When visualized as a position vector starting from the origin, it would move right by 0.1 units and up by 0.2 units.

In summary, the component form of a vector provides a clear numerical representation of its direction and magnitude in two-dimensional space. By understanding how to derive and visualize position vectors, students can effectively analyze and manipulate vectors in various mathematical contexts.

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Problem

True or false: If $a ⃗ equals left angle bracket 3 comma 2 right angle bracket$ and $b ⃗$ has initial point $(3,-1)$ & terminal point $(6,1)$ , then $a ⃗=b ⃗$ .

True

False

Cannot be determined with given information

Problem

True or false: If $a ⃗ = ⟨ 3, - 2 ⟩$ and $b ⃗$ has initial point $(-10,5)$ & terminal point $(-7,3)$ , then $a ⃗=b ⃗$ .

True

False

Cannot be determined with the given information

Example

Position Vectors & Component Form Example 1

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Position Vectors & Component Form Example 1 Video Summary

To find the vector v in component form given its initial point (4, 3) and terminal point (1, 2), we can use the formula for vector components:

v = (x₂ - x₁, y₂ - y₁)

Here, (x₁, y₁) represents the initial point and (x₂, y₂) represents the terminal point. Substituting the values:

x₂ = 1, x₁ = 4, y₂ = 2, y₁ = 3

We calculate the components:

v = (1 - 4, 2 - 3)

Performing the subtraction gives:

v = (-3, -1)

This means the vector v in component form is v = -3i - 1j, where i and j are the unit vectors in the x and y directions, respectively.

Next, to sketch the position vector, we start at the origin (0, 0). From the origin, we move 3 units to the left (negative x-direction) and 1 unit down (negative y-direction). This results in the endpoint of the vector being at the coordinates (-3, -1). The sketch will show a vector pointing from the origin to the point (-3, -1), illustrating the negative components of the vector.

In summary, the vector v in component form is v = (-3, -1), and its position vector can be visualized starting from the origin and extending to the point (-3, -1) in the Cartesian plane.

Concept

Finding Magnitude of a Vector

Video duration:

Finding Magnitude of a Vector Video Summary

Understanding how to find the magnitude of a vector is essential in vector analysis. The magnitude represents the total length of the vector, which can be calculated using the components of the vector in component form. A vector in component form is expressed as (x, y), where x and y are the horizontal and vertical components, respectively.

To find the magnitude of a vector, we can apply the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. This can be expressed mathematically as:

c^2 = a^2 + b^2

Rearranging this gives us:

c = \sqrt{a^2 + b^2}

In the context of vectors, the components (x, y) can be treated as the legs of a right triangle, where x is one leg and y is the other. Therefore, the magnitude |v| of a vector can be calculated using the formula:

|v| = \sqrt{x^2 + y^2}

For example, if we have a vector with components (4, 3), the magnitude would be:

|v| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5

In another scenario, if we need to find the magnitude of a vector defined by two points, say point P at (1, 2) and point Q at (5, 3), we first determine the vector in component form. The components can be calculated as:

v = (x2 - x1, y2 - y1) = (5 - 1, 3 - 2) = (4, 1)

Now, we can find the magnitude:

|v| = \sqrt{4^2 + 1^2} = \sqrt{16 + 1} = \sqrt{17}

This process illustrates that whether given directly in component form or derived from points, the magnitude of a vector can always be calculated using the same fundamental principles. Mastering this concept is crucial for further studies in physics and engineering, where vector analysis plays a significant role.

Problem

If vector $v ⃗ = ⟨ - 4, - 10 ⟩$ , calculate the magnitude $|v ⃗ |$ .

$4$\surd$29$

$2$\surd$116$

$2√29$

$$\surd$-116$

Problem

If vector $v ⃗$ has initial point $(-1,2)$ and terminal point $(9,5)$ , calculate the magnitude $|v ⃗ |$ .

$2√29$

$$\surd$109$

$$\surd$73$

$$\surd$113$

Example

Finding Magnitude of a Vector Example 2

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Finding Magnitude of a Vector Example 2 Video Summary

To determine the vector $ \mathbf{v} $ given its initial point $ (1, 2) $ and terminal point $ (4, 4) $, we first express the vector in component form. The components of the vector can be calculated by subtracting the coordinates of the initial point from those of the terminal point. Specifically, the x-component $ v_x $ is calculated as:

\[ v_x = x_2 - x_1 = 4 - 1 = 3 \]

Similarly, the y-component $ v_y $ is calculated as:

\[ v_y = y_2 - y_1 = 4 - 2 = 2 \]

Thus, the vector $ \mathbf{v} $ in component form is $ \mathbf{v} = (3, 2) $.

Next, we can sketch the position vector, which starts at the origin $ (0, 0) $. To do this, we move 3 units along the x-axis and 2 units along the y-axis, resulting in the endpoint at $ (3, 2) $. This visual representation helps in understanding the direction and magnitude of the vector.

To find the magnitude of vector $ \mathbf{v} $, we use the formula for the magnitude of a vector:

\[ \|\mathbf{v}\| = \sqrt{v_x^2 + v_y^2} \]

Substituting the values of $ v_x $ and $ v_y $ into the equation gives:

\[ \|\mathbf{v}\| = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13} \]

Therefore, the magnitude of vector $ \mathbf{v} $ is $ \sqrt{13} $. This concludes the calculation of the vector's component form, its sketch as a position vector, and its magnitude.

Concept

Algebraic Operations on Vectors

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Algebraic Operations on Vectors Video Summary

In vector mathematics, understanding how to add and subtract vectors is crucial, especially when dealing with their component forms. The tip-to-tail method is a visual approach where the tip of one vector is placed at the tail of another, and the resultant vector connects the starting point of the first vector to the endpoint of the second. This method can be complemented by a numerical approach, which involves manipulating the individual components of the vectors.

When given two vectors in component form, such as vector v and vector u, you can add or subtract them by focusing on their x and y components. For example, if v is represented as (v_x, v_y) and u as (u_x, u_y), the sum or difference of these vectors can be calculated as follows:

For addition: v + u = (v_x + u_x, v_y + u_y)

For subtraction: v - u = (v_x - u_x, v_y - u_y)

For instance, if v is (2, 3) and u is (3, -1), the addition would yield:

v + u = (2 + 3, 3 + (-1)) = (5, 2)

This result can be verified visually using the tip-to-tail method, confirming that the mathematical approach aligns with the graphical representation.

Another important operation is multiplying a vector by a scalar. To multiply a vector by a scalar, you distribute the scalar to each component of the vector. If u is (2, 4) and you want to find 3u, you would calculate:

3u = (3 * 2, 3 * 4) = (6, 12)

To find the vector v - 3u, where v is (8, 5), you would perform the following steps:

1. Calculate 3u as shown above.

2. Subtract the components: v - 3u = (8 - 6, 5 - 12) = (2, -7)

This process illustrates how to perform basic vector operations using numerical values rather than graphical representations, which is a vital skill in both mathematics and science courses.

Problem

If vectors $v ⃗ equals left angle bracket 2 comma 1 right angle bracket$ , $u ⃗ equals left angle bracket 3 comma 4 right angle bracket$ , and $w ⃗ = ⟨ - 1, 1 ⟩$ , calculate $v ⃗ + u ⃗ - w ⃗$ .

$left angle bracket 4 comma 4 right angle bracket$

$left angle bracket 5 comma 4 right angle bracket$

$left angle bracket 6 comma 6 right angle bracket$

$left angle bracket 6 comma 4 right angle bracket$

Problem

If vectors $v ⃗ equals left angle bracket 4 comma 1 right angle bracket$ , $u ⃗=⟨-8,3⟩$ , and $w ⃗=⟨-2,-1⟩$ , calculate $w ⃗-3(v ⃗+u ⃗)$ .

$\langle\)10,13\(\rangle$

$\langle\)-14,13\(\rangle$

$\langle\)-14,-13\(\rangle$

$\langle\)10,-13\(\rangle$

Example

Algebraic Operations on Vectors Example 1

Video duration:

Algebraic Operations on Vectors Example 1 Video Summary

To solve for the magnitude of vector c, defined as c = 5a - 2b, we start with the given vectors: a = (3, 1) and b = (-4, 9). The first step is to calculate 5a and 2b.

Calculating 5a involves multiplying each component of vector a by 5:

5a = 5 * (3, 1) = (5 * 3, 5 * 1) = (15, 5).

Next, we calculate 2b by multiplying each component of vector b by 2:

2b = 2 * (-4, 9) = (2 * -4, 2 * 9) = (-8, 18).

Now, we can find vector c by subtracting 2b from 5a:

c = 5a - 2b = (15, 5) - (-8, 18) = (15 + 8, 5 - 18) = (23, -13).

With vector c determined as (23, -13), we can now find its magnitude using the Pythagorean theorem:

|c| = √(x_c² + y_c²), where x_c = 23 and y_c = -13.

Calculating the squares:

23² = 529 and (-13)² = 169.

Adding these results gives:

529 + 169 = 698.

Thus, the magnitude of vector c is:

|c| = √698.

This process illustrates how to handle multiple vector operations and find the magnitude of a resultant vector effectively.

Your Algebra and Trigonometry tutors

Patrick Ford

Physics and Math Lead Instructor

Vectors in Component Form : Videos & Practice Problems

Position Vectors & Component Form

Position Vectors & Component Form Video Summary

True or false: If a⃗=⟨3,2⟩a ⃗=⟨3,2⟩ and b⃗b ⃗b⃗ has initial point (3,−1)(3,-1)(3,−1) & terminal point (6,1)(6,1)(6,1), then a⃗=b⃗a ⃗=b ⃗a⃗=b⃗.

True or false: If a⃗=⟨3,−2⟩a ⃗=⟨3,-2⟩ and b⃗b ⃗b⃗ has initial point (−10,5)(-10,5)(−10,5) & terminal point (−7,3)(-7,3)(−7,3), then a⃗=b⃗a ⃗=b ⃗a⃗=b⃗.

Position Vectors & Component Form Example 1

Position Vectors & Component Form Example 1 Video Summary

Finding Magnitude of a Vector

Finding Magnitude of a Vector Video Summary

If vector v⃗=⟨−4,−10⟩v ⃗=⟨-4,-10⟩, calculate the magnitude ∣v⃗∣|v ⃗ |∣v⃗∣.

If vector v⃗v ⃗ has initial point (−1,2)(-1,2)(−1,2) and terminal point (9,5)(9,5)(9,5), calculate the magnitude ∣v⃗∣|v ⃗ |∣v⃗∣.

Finding Magnitude of a Vector Example 2

Finding Magnitude of a Vector Example 2 Video Summary

Algebraic Operations on Vectors

Algebraic Operations on Vectors Video Summary

If vectors v⃗=⟨2,1⟩v ⃗=⟨2,1⟩, u⃗=⟨3,4⟩u ⃗=⟨3,4⟩, and w⃗=⟨−1,1⟩w ⃗=⟨-1,1⟩, calculate v⃗+u⃗−w⃗v ⃗+u ⃗-w ⃗.

If vectors v⃗=⟨4,1⟩v ⃗=⟨4,1⟩, u⃗=⟨−8,3⟩u ⃗=⟨-8,3⟩u⃗=⟨−8,3⟩, and w⃗=⟨−2,−1⟩w ⃗=⟨-2,-1⟩w⃗=⟨−2,−1⟩, calculate w⃗−3(v⃗+u⃗)w ⃗-3(v ⃗+u ⃗)w⃗−3(v⃗+u⃗).

Algebraic Operations on Vectors Example 1

Algebraic Operations on Vectors Example 1 Video Summary

Your Algebra and Trigonometry tutors

True or false: If $a ⃗ equals left angle bracket 3 comma 2 right angle bracket$ and $b ⃗$ has initial point $(3,-1)$ & terminal point $(6,1)$ , then $a ⃗=b ⃗$ .

True or false: If $a ⃗ = ⟨ 3, - 2 ⟩$ and $b ⃗$ has initial point $(-10,5)$ & terminal point $(-7,3)$ , then $a ⃗=b ⃗$ .

If vector $v ⃗ = ⟨ - 4, - 10 ⟩$ , calculate the magnitude $|v ⃗ |$ .

If vector $v ⃗$ has initial point $(-1,2)$ and terminal point $(9,5)$ , calculate the magnitude $|v ⃗ |$ .

If vectors $v ⃗ equals left angle bracket 2 comma 1 right angle bracket$ , $u ⃗ equals left angle bracket 3 comma 4 right angle bracket$ , and $w ⃗ = ⟨ - 1, 1 ⟩$ , calculate $v ⃗ + u ⃗ - w ⃗$ .

If vectors $v ⃗ equals left angle bracket 4 comma 1 right angle bracket$ , $u ⃗=⟨-8,3⟩$ , and $w ⃗=⟨-2,-1⟩$ , calculate $w ⃗-3(v ⃗+u ⃗)$ .