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

√ − 116 \surd-116 √ − 116

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

⟨ 6 , 4 ⟩ \langle6,4\rangle ⟨ 6 , 4 ⟩

⟨ 4 , 4 ⟩ \langle4,4\rangle ⟨ 4 , 4 ⟩

⟨ 5 , 4 ⟩ \langle5,4\rangle ⟨ 5 , 4 ⟩

⟨ 6 , 6 ⟩ \langle6,6\rangle ⟨ 6 , 6 ⟩

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

⟨ 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 ⟩

14. Vectors

Vectors in Component Form

14. Vectors

Vectors in Component Form: Videos & Practice Problems

Video Lessons Practice Worksheet

Topic summary

Understanding vectors involves representing them in component form, which consists of x and y components indicating their lengths in respective directions. The magnitude of a vector can be calculated using the Pythagorean theorem, where the magnitude is the square root of the sum of the squares of its components. Additionally, vectors can be added or subtracted by combining their individual components. This foundational knowledge is crucial for further studies in mathematics and physics, enabling students to manipulate and analyze vector quantities effectively.

concept

Position Vectors & Component Form

Video duration:

Position Vectors & Component Form Video Summary

Vectors are fundamental mathematical objects that can be visually represented as arrows in space, indicating both direction and magnitude. A key concept in vector analysis is the position vector, which is defined as a vector whose initial point is located at the origin of a coordinate system. This allows for a standardized way to express vectors in a two-dimensional space.

To express a vector in component form, we break it down into its x and y components. For instance, if a vector v moves 3 units to the right and 2 units up, it can be represented as v = (3, 2). This notation succinctly captures the vector's movement in both the horizontal (x) and vertical (y) directions.

When given two points, the initial point and the terminal point of a vector, we can calculate the vector's component form using the formula:

v = (x₂ - x₁, y₂ - y₁)

Here, (x₁, y₁) represents the initial point and (x₂, y₂) represents the terminal point. For example, if the initial point is (0.2, 0.3) and the terminal point is (0.3, 0.5), we can find the vector v as follows:

v = (0.3 - 0.2, 0.5 - 0.3) = (0.1, 0.2)

This calculation shows that the vector moves 0.1 units in the x direction and 0.2 units in the y direction. When visualizing this vector as a position vector starting from the origin, it would extend 0.1 units to the right and 0.2 units up.

Understanding how to represent vectors in component form is crucial for solving various problems in physics and engineering, where vectors are used to describe forces, velocities, and other directional quantities. By mastering these concepts, students can effectively analyze and manipulate vectors in both theoretical and practical applications.

Problem

True or false: If $a ⃗=⟨3,2⟩$ 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

Video duration:

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 = (-3, -1).

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 point (-3, -1). The sketch will show a vector pointing from the origin to the point (-3, -1), illustrating that the vector moves towards the negative side of both the x and y axes.

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

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 and can be calculated using the components of the vector in component form, typically denoted as (x, y).

To find the magnitude of a vector, we can apply the Pythagorean theorem, which states that for a right triangle, the relationship between the lengths of the sides is given by the equation:

a² + b² = c²

Here, c represents the hypotenuse, which corresponds to the magnitude of the vector, while a and b are the lengths of the other two sides. Rearranging this gives us:

c = \sqrt{a² + b²}

In the context of vectors, if we have a vector represented as (x, y), the magnitude can be calculated using:

||v|| = \sqrt{x² + y²}

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

||v|| = \sqrt{4² + 3²} = \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:

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

Now, we can calculate the magnitude:

||v|| = \sqrt{4² + 1²} = \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 $|v ⃗ |$ .

$4\surd29$

$2\surd116$

$2√29$

$\surd-116$

Problem

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

$2√29$

$\surd109$

$\surd73$

$\surd113$

example

Finding Magnitude of a Vector Example 1

Video duration:

Finding Magnitude of a Vector Example 1 Video Summary

To determine the vector $ \mathbf{v} $ given its initial point $ (1, 2) $ and terminal point $ (4, 4) $, we first convert it into 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 represented as:

$ \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) $.

Finally, 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 we calculated:

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

Therefore, the magnitude of vector $ \mathbf{v} $ is $ \sqrt{13} $. This encapsulates the component form of the vector, its graphical representation as a position vector, and the calculation of 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 = (2, 3) and u = (3, -1), the addition would yield:

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

This result can also be verified visually using the tip-to-tail method, confirming the mathematical operations align with graphical representations.

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 = (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 = (8, 5), you would perform the following steps:

1. Calculate 3u = (6, 12).

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

This process illustrates how to effectively perform basic vector operations using both graphical and numerical methods, which is essential for success in mathematics and science courses.

Problem

If vectors $v ⃗=⟨2,1⟩$ , $u ⃗=⟨3,4⟩$ , and $w ⃗=⟨-1,1⟩$ , calculate $v ⃗+u ⃗-w ⃗$ .

$\langle4,4\rangle$

$\langle5,4\rangle$

$\langle6,6\rangle$

$\langle6,4\rangle$

Problem

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

$\langle10,13\rangle$

$\langle-14,13\rangle$

$\langle-14,-13\rangle$

$\langle10,-13\rangle$

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.

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Vectors in Component Form

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Vectors in Component Form definitions
14. Vectors
15 Terms

Here’s what students ask on this topic:

What is the component form of a vector?

The component form of a vector is a way to represent the vector using its horizontal (x) and vertical (y) components. If a vector v has an initial point at the origin and a terminal point at (x, y), then the component form of the vector is written as v = (x, y). This notation indicates that the vector moves x units in the horizontal direction and y units in the vertical direction. For example, a vector that moves 3 units to the right and 2 units up is represented as (3, 2).

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How do you find the magnitude of a vector in component form?

To find the magnitude of a vector in component form, you use the Pythagorean theorem. If a vector v has components (x, y), the magnitude |v| is calculated as:

$\sqrt{x^{2} + y^{2}}$

For example, if the vector is (4, 3), the magnitude is:

$\sqrt{42 + 32} = \sqrt{16 + 9} = \sqrt{25} = 5$

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How do you add or subtract vectors in component form?

To add or subtract vectors in component form, you simply add or subtract their corresponding components. If you have vectors v = (x₁, y₁) and u = (x₂, y₂), the sum v + u is:

(x₁ + x₂, y₁ + y₂)

Similarly, the difference v - u is:

(x₁ - x₂, y₁ - y₂)

For example, if v = (2, 3) and u = (3, -1), then:

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

v - u = (2 - 3, 3 + 1) = (-1, 4)

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How do you multiply a vector by a scalar in component form?

To multiply a vector by a scalar, you multiply each component of the vector by the scalar. If vector v = (x, y) and scalar k, then the product kv is:

(k * x, k * y)

For example, if v = (2, 4) and k = 3, then:

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

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How do you find the component form of a vector given its initial and terminal points?

To find the component form of a vector given its initial point (x₁, y₁) and terminal point (x₂, y₂), you subtract the coordinates of the initial point from the coordinates of the terminal point. The component form is:

(x₂ - x₁, y₂ - y₁)

For example, if the initial point is (2, 3) and the terminal point is (5, 7), then the component form is:

(5 - 2, 7 - 3) = (3, 4)

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Your Precalculus 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⟩a⃗=⟨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⟩a⃗=⟨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⟩v⃗=⟨−4,−10⟩, calculate the magnitude ∣v⃗∣|v ⃗ |∣v⃗∣.

If vector v⃗v ⃗ v⃗ 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 1

Finding Magnitude of a Vector Example 1 Video Summary

Algebraic Operations on Vectors

Algebraic Operations on Vectors Video Summary

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

If vectors v⃗=⟨4,1⟩v ⃗=⟨4,1⟩v⃗=⟨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

Do you want more practice?

Go over this topic definitions with flashcards

Here’s what students ask on this topic:

What is the component form of a vector?

How do you find the magnitude of a vector in component form?

How do you add or subtract vectors in component form?

How do you multiply a vector by a scalar in component form?

How do you find the component form of a vector given its initial and terminal points?

Your Precalculus tutors

True or false: If $a ⃗=⟨3,2⟩$ 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 $v ⃗=⟨2,1⟩$ , $u ⃗=⟨3,4⟩$ , and $w ⃗=⟨-1,1⟩$ , calculate $v ⃗+u ⃗-w ⃗$ .

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