If vector v ⃗ = 12 ı ^ − 2 ȷ ^ v⃗=12î-2ĵ v ⃗ = 12 ı ^ − 2 ȷ ^ and vector u ⃗ = 5 ı ^ + 20 ȷ ^ u⃗=5î+20ĵ u ⃗ = 5 ı ^ + 20 ȷ ^ calculate 2 v ⃗ − 2 u ⃗ 2v ⃗-2u ⃗ 2 v ⃗ − 2 u ⃗ using ı ^ î ı ^ and ȷ ^ ĵ ȷ ^ notation.

14 ı ^ − 44 ȷ ^ 14î-44ĵ 14 ı ^ − 44 ȷ ^

7 ı ^ − 18 ȷ ^ 7î-18ĵ 7 ı ^ − 18 ȷ ^

14 ı ^ + 44 ȷ ^ 14î+44ĵ 14 ı ^ + 44 ȷ ^

14 ı ^ − 36 ȷ ^ 14î-36ĵ 14 ı ^ − 36 ȷ ^

If vector v ⃗ = v ⃗= v ⃗ = 11 ȷ ^ 11ĵ 11 ȷ ^ and vector u ⃗ = u⃗= u ⃗ = 10 ı ^ − 25 ȷ ^ 10î-25ĵ 10 ı ^ − 25 ȷ ^ calculate v ⃗ + 1 5 u ⃗ v⃗+\frac15u⃗ v ⃗ + 5 1 u ⃗ using ı ^ î ı ^ & ȷ ^ ĵ ȷ ^ notation.

2 ı ^ + 6 ȷ ^ 2î+6ĵ 2 ı ^ + 6 ȷ ^

13 ı ^ + 14 ȷ ^ 13î+14ĵ 13 ı ^ + 14 ȷ ^

2 ı ^ + 14 ȷ ^ 2î+14ĵ 2 ı ^ + 14 ȷ ^

13 ı ^ − 5 ȷ ^ 13î-5ĵ 13 ı ^ − 5 ȷ ^

If vector a ⃗ = 20 ı ^ a⃗=20î a ⃗ = 20 ı ^ and vector b ⃗ = 50 ȷ ^ b⃗=50ĵ b ⃗ = 50 ȷ ^ calculate a ⃗ − b ⃗ a ⃗-b ⃗ a ⃗ − b ⃗ using ı ^ î ı ^ and ȷ ^ ĵ ȷ ^ notation.

20 ı ^ − 50 ȷ ^ 20î-50ĵ 20 ı ^ − 50 ȷ ^

20 ı ^ + 50 ȷ ^ 20î+50ĵ 20 ı ^ + 50 ȷ ^

Find the unit vector in the direction of a ⃗ = 6 ı ^ + 3 ȷ ^ a⃗=6î+3ĵ a ⃗ = 6 ı ^ + 3 ȷ ^ .

a ^ = 2 √ 5 5 ı ^ + √ 5 5 ȷ ^ â=\frac{2\surd5}{5}î+\frac{\surd5}{5}ĵ a ^ = 5 2√5 ı ^ + 5 √5 ȷ ^

a ^ = 3 √ 5 ȷ ^ â=3\surd5ĵ a ^ = 3√5 ȷ ^

a ^ = 2 √ 5 5 ı ^ − √ 5 5 ȷ ^ â=\frac{2\surd5}{5}î-\frac{\surd5}{5}ĵ a ^ = 5 2√5 ı ^ − 5 √5 ȷ ^

a ^ = 2 √ 5 5 ı ^ + 3 √ 5 5 ȷ ^ â=\frac{2\surd5}{5}î+\frac{3\surd5}{5}ĵ a ^ = 5 2√5 ı ^ + 5 3√5 ȷ ^

Find the unit vector in the direction of v ⃗ = 12 ı ^ − 35 ȷ ^ v⃗=12î-35ĵ v ⃗ = 12 ı ^ − 35 ȷ ^ .

v ^ = 12 37 ı ^ − 35 37 ȷ ^ v̂=\frac{12}{37}î-\frac{35}{37}ĵ v ^ = 37 12 ı ^ − 37 35 ȷ ^

v ^ = 37 ı ^ ^ v̂=37î̂ v ^ = 37 ı ^ ^

v ^ = 35 ı ^ − 12 ȷ ^ v̂=35î-12ĵ v ^ = 35 ı ^ − 12 ȷ ^

v ^ = 35 37 ı ^ − 12 37 ȷ ^ v̂=\frac{35}{37}î-\frac{12}{37}ĵ v ^ = 37 35 ı ^ − 37 12 ȷ ^

8. Vectors

Unit Vectors and i & j Notation

8. Vectors

Unit Vectors and i & j Notation: Videos & Practice Problems

Video Lessons Practice Worksheet

Topic summary

Vectors can be represented using I and j notation, where I hat and j hat are unit vectors in the x and y directions, respectively. To convert a vector into this notation, identify its components and express it as a combination of I and j. For any vector, the unit vector can be calculated by dividing the vector by its magnitude, ensuring it has a magnitude of 1. This method is essential for understanding vector operations and simplifies calculations in various applications.

concept

i & j Notation

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i & j Notation Video Summary

In the study of vectors, understanding how to represent them using I and j notation is essential. This notation simplifies the representation of vectors by breaking them down into their components along the x and y axes. A unit vector is a vector with a magnitude of 1 that indicates direction, and the unit vectors in the x and y directions are denoted as I hat (for the x direction) and j hat (for the y direction). These can also be referred to as x hat and y hat in some texts, but they convey the same concept.

To express a vector in I and j notation, you identify how many unit vectors are needed in each direction. For example, if you have a vector v that requires 4 unit vectors in the x direction and 3 in the y direction, it can be represented as:

v = 4I + 3j

When given a vector in component form, such as (4, 3), you can easily convert it to I and j notation by multiplying the x component by I hat and the y component by j hat. Thus, (4, 3) becomes:

4I + 3j

Using this notation allows for straightforward vector operations. For instance, if you have two vectors, u = (2, 4) and v = (1, 0), their representations in I and j notation would be:

u = 2I + 4j

v = 1I + 0j = I

To add these vectors, you simply combine their components:

u + v = (2I + 4j) + (I) = 3I + 4j

For subtraction, such as finding u - 2v, you first calculate 2v:

2v = 2I

Then, perform the subtraction:

u - 2v = (2I + 4j) - (2I) = 0 + 4j = 4j

This illustrates how I and j notation can simplify vector operations, making it easier to visualize and compute with vectors in two-dimensional space.

Problem

If vector $v⃗=12î-2ĵ$ and vector $u⃗=5î+20ĵ$ calculate $2v ⃗-2u ⃗$ using $î$ and $ĵ$ notation.

$14î-44ĵ$

$7î-18ĵ$

$14î+44ĵ$

$14î-36ĵ$

Problem

If vector $v ⃗=$ $11ĵ$ and vector $u⃗=$ $10î-25ĵ$ calculate $v⃗+\frac15u⃗$ using $î$ & $ĵ$ notation.

$13î+14ĵ$

$2î+14ĵ$

$13î-5ĵ$

$2î+6ĵ$

Problem

If vector $a⃗=20î$ and vector $b⃗=50ĵ$ calculate $a ⃗-b ⃗$ using $î$ and $ĵ$ notation.

$-30ĵ$

$20î-50ĵ$

$30î̂$

$20î+50ĵ$

concept

Unit Vector in the Direction of a Given Vector

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Unit Vector in the Direction of a Given Vector Video Summary

In the study of vectors, understanding how to calculate unit vectors is essential, especially when dealing with vectors that point in various directions. A unit vector is defined as a vector that has a magnitude of 1 and indicates direction. To find a unit vector that points in the same direction as a given vector, you can follow a straightforward process.

Consider a vector v represented as v = 4i + 3j. To determine the corresponding unit vector, denoted as v̂, you first need to calculate the magnitude of the vector v. The magnitude is calculated using the formula:

|v| = √(x² + y²)

In this case, the x-component is 4 and the y-component is 3. Thus, the magnitude becomes:

|v| = √(4² + 3²) = √(16 + 9) = √25 = 5

With the magnitude known, you can find the unit vector by dividing each component of the vector by its magnitude:

v̂ = (4/5)i + (3/5)j

This unit vector v̂ retains the direction of the original vector v but has a magnitude of 1. To verify that v̂ is indeed a unit vector, you can calculate its magnitude using the same method:

|v̂| = √((4/5)² + (3/5)²) = √(16/25 + 9/25) = √(25/25) = √1 = 1

Since the magnitude of v̂ equals 1, it confirms that v̂ is a unit vector. This process illustrates that to find a unit vector in any direction, simply divide the vector by its magnitude. This method not only provides the unit vector but also serves as a check to ensure accuracy in calculations.

example

Unit Vector in the Direction of a Given Vector Example 1

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Unit Vector in the Direction of a Given Vector Example 1 Video Summary

To find the unit vector $ \hat{u} $ in the direction of a given vector $ \mathbf{u} $, we use the formula:

$ \hat{u} = \frac{\mathbf{u}}{|\mathbf{u}|} $

In this case, the vector $ \mathbf{u} $ is given as $ -\mathbf{i} + 2\mathbf{j} $, which can also be expressed as $ -1\mathbf{i} + 2\mathbf{j} $. To proceed, we first need to calculate the magnitude of vector $ \mathbf{u} $, denoted as $ |\mathbf{u}| $.

The magnitude of a vector is calculated using the formula:

$ |\mathbf{u}| = \sqrt{x^2 + y^2} $

Here, the $ x $ component is $ -1 $ and the $ y $ component is $ 2 $. Thus, we compute:

$ |\mathbf{u}| = \sqrt{(-1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5} $

Now that we have the magnitude, we can substitute it back into the unit vector formula:

$ \hat{u} = \frac{-1\mathbf{i} + 2\mathbf{j}}{\sqrt{5}} $

This results in:

$ \hat{u} = \left(-\frac{1}{\sqrt{5}}\right)\mathbf{i} + \left(\frac{2}{\sqrt{5}}\right)\mathbf{j} $

To simplify the expression and eliminate the square root from the denominator, we rationalize it by multiplying the numerator and denominator by $ \sqrt{5} $:

$ \hat{u} = \left(-\frac{1 \cdot \sqrt{5}}{5}\right)\mathbf{i} + \left(\frac{2 \cdot \sqrt{5}}{5}\right)\mathbf{j} $

Thus, the final form of the unit vector $ \hat{u} $ is:

$ \hat{u} = -\frac{\sqrt{5}}{5}\mathbf{i} + \frac{2\sqrt{5}}{5}\mathbf{j} $

This unit vector represents the direction of vector $ \mathbf{u} $ with a magnitude of 1, making it useful for various applications in physics and engineering.

Problem

Find the unit vector in the direction of $a⃗=6î+3ĵ$ .

$â=3\surd5ĵ$

$â=\frac{2\surd5}{5}î-\frac{\surd5}{5}ĵ$

$â=\frac{2\surd5}{5}î+\frac{\surd5}{5}ĵ$

$â=\frac{2\surd5}{5}î+\frac{3\surd5}{5}ĵ$

Problem

Find the unit vector in the direction of $v⃗=12î-35ĵ$ .

$v̂=\frac{12}{37}î-\frac{35}{37}ĵ$

$v̂=37î̂$

$v̂=35î-12ĵ$

$v̂=\frac{35}{37}î-\frac{12}{37}ĵ$

example

Unit Vector in the Direction of a Given Vector Example 2

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Unit Vector in the Direction of a Given Vector Example 2 Video Summary

To find the unit vector in the direction of vector c, we start with the given vectors: a = (-1, 3) and b = (4, 7). The vector c is defined as:

c = 4a + 2b.

First, we calculate vector c by scaling the vectors a and b:

4a = 4 * (-1, 3) = (-4, 12)

2b = 2 * (4, 7) = (8, 14)

Next, we add the corresponding components of these scaled vectors:

c = (-4 + 8, 12 + 14) = (4, 26).

Now that we have vector c, we need to find its magnitude. The magnitude of a vector is calculated using the formula:

|c| = √(x² + y²),

where x and y are the components of the vector. For c = (4, 26), we compute:

|c| = √(4² + 26²) = √(16 + 676) = √(692).

To simplify √(692), we factor it as:

√(692) = √(4 * 173) = √(4) * √(173) = 2√(173).

With the magnitude of vector c determined, we can now find the unit vector, denoted as ĉ. The unit vector is calculated by dividing each component of c by its magnitude:

ĉ = c / |c| = (4, 26) / (2√(173)) = (4 / (2√(173)), 26 / (2√(173))).

This simplifies to:

ĉ = (2 / √(173), 13 / √(173)).

To express the unit vector in a more standard form, we rationalize the denominators by multiplying the numerator and denominator by √(173):

ĉ = (2√(173) / 173, 13√(173) / 173).

This final expression represents the unit vector in the direction of c:

ĉ = (2√(173) / 173, 13√(173) / 173).

By following these steps—calculating the vector, finding its magnitude, and then deriving the unit vector—you can effectively solve similar problems involving vectors and their directions.

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Here’s what students ask on this topic:

What is i and j notation in vectors?

i and j notation is a way to represent vectors using unit vectors. The unit vector $i_{^}$ points in the x-direction, and the unit vector $j_{^}$ points in the y-direction. Any vector can be expressed as a combination of these unit vectors. For example, a vector with components (4, 3) can be written as $4 i_{^} + 3 j_{^}$ . This notation simplifies vector operations and calculations.

How do you convert a vector to i and j notation?

To convert a vector to i and j notation, identify its x and y components. Multiply the x component by the unit vector $i_{^}$ and the y component by the unit vector $j_{^}$ . For example, a vector with components (2, 4) can be written as $2 i_{^} + 4 j_{^}$ . This method helps in visualizing and performing vector operations more easily.

How do you find a unit vector in the same direction as a given vector?

To find a unit vector in the same direction as a given vector, divide the vector by its magnitude. The magnitude of a vector $(a, b)$ is calculated as $\sqrt{a^{2} + b^{2}}$ . For example, for vector $(4, 3)$ , the magnitude is 5. The unit vector is $(\frac{4}{5}, \frac{3}{5})$ , or $\frac{4}{5} i_{^} + \frac{3}{5} j_{^}$ .

What is the difference between component form and i and j notation?

Component form represents a vector as an ordered pair (x, y), showing its x and y components directly. i and j notation, on the other hand, expresses the vector as a combination of unit vectors $i_{^}$ and $j_{^}$ . For example, a vector with components (3, 2) in component form is written as $(3, 2)$ , while in i and j notation, it is $3 i_{^} + 2 j_{^}$ . Both forms convey the same information but are used in different contexts for convenience.

How do you add vectors using i and j notation?

To add vectors using i and j notation, add their respective components. For example, if vector $u = 2 i_{^} + 4 j_{^}$ and vector $v = 1 i_{^}$ , then $u + v = (2 + 1) i_{^} + (4 + 0) j_{^}$ , which simplifies to $3 i_{^} + 4 j_{^}$ . This method is straightforward and aligns with component-wise addition.

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Patrick Ford

Physics and Math Lead Instructor

Unit Vectors and i & j Notation: Videos & Practice Problems

i & j Notation

i & j Notation Video Summary

If vector v⃗=12ı^−2ȷ^v⃗=12î-2ĵv⃗=12ı^−2ȷ^ and vector u⃗=5ı^+20ȷ^u⃗=5î+20ĵu⃗=5ı^+20ȷ^ calculate 2v⃗−2u⃗2v ⃗-2u ⃗2v⃗−2u⃗ using ı^îı^ and ȷ^ĵȷ^ notation.

If vector v⃗=v ⃗=v⃗= 11ȷ^11ĵ11ȷ^ and vector u⃗=u⃗=u⃗= 10ı^−25ȷ^10î-25ĵ10ı^−25ȷ^ calculate v⃗+15u⃗v⃗+\frac15u⃗v⃗+51u⃗ using ı^îı^ & ȷ^ĵȷ^ notation.

If vector a⃗=20ı^a⃗=20îa⃗=20ı^ and vector b⃗=50ȷ^b⃗=50ĵb⃗=50ȷ^ calculate a⃗−b⃗a ⃗-b ⃗a⃗−b⃗ using ı^îı^ and ȷ^ĵȷ^ notation.

Unit Vector in the Direction of a Given Vector

Unit Vector in the Direction of a Given Vector Video Summary

Unit Vector in the Direction of a Given Vector Example 1

Unit Vector in the Direction of a Given Vector Example 1 Video Summary

Find the unit vector in the direction of a⃗=6ı^+3ȷ^a⃗=6î+3ĵa⃗=6ı^+3ȷ^.

Find the unit vector in the direction of v⃗=12ı^−35ȷ^v⃗=12î-35ĵv⃗=12ı^−35ȷ^.

Unit Vector in the Direction of a Given Vector Example 2

Unit Vector in the Direction of a Given Vector Example 2 Video Summary

Do you want more practice?

Here’s what students ask on this topic:

What is i and j notation in vectors?

How do you convert a vector to i and j notation?

How do you find a unit vector in the same direction as a given vector?

What is the difference between component form and i and j notation?

How do you add vectors using i and j notation?

Your Trigonometry tutors

If vector $v⃗=12î-2ĵ$ and vector $u⃗=5î+20ĵ$ calculate $2v ⃗-2u ⃗$ using $î$ and $ĵ$ notation.

If vector $v ⃗=$ $11ĵ$ and vector $u⃗=$ $10î-25ĵ$ calculate $v⃗+\frac15u⃗$ using $î$ & $ĵ$ notation.

If vector $a⃗=20î$ and vector $b⃗=50ĵ$ calculate $a ⃗-b ⃗$ using $î$ and $ĵ$ notation.

Find the unit vector in the direction of $a⃗=6î+3ĵ$ .

Find the unit vector in the direction of $v⃗=12î-35ĵ$ .