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

14ı^−44ȷ^14î-44ĵ14ı^−44ȷ^

7ı^−18ȷ^7î-18ĵ7ı^−18ȷ^

14ı^+44ȷ^14î+44ĵ14ı^+44ȷ^

14ı^−36ȷ^14î-36ĵ14ı^−36ȷ^

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

13ı^+14ȷ^13î+14ĵ13ı^+14ȷ^

13ı^−5ȷ^13î-5ĵ13ı^−5ȷ^

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

20ı^−50ȷ^20î-50ĵ20ı^−50ȷ^

20ı^+50ȷ^20î+50ĵ20ı^+50ȷ^

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

a^=2√55ı^+√55ȷ^â=\frac{2\surd5}{5}î+\frac{\surd5}{5}ĵa^=52√5ı^+5√5ȷ^

a^=3√5ȷ^â=3\surd5ĵa^=3√5ȷ^

a^=2√55ı^−√55ȷ^â=\frac{2\surd5}{5}î-\frac{\surd5}{5}ĵa^=52√5ı^−5√5ȷ^

a^=2√55ı^+3√55ȷ^â=\frac{2\surd5}{5}î+\frac{3\surd5}{5}ĵa^=52√5ı^+53√5ȷ^

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

v^=1237ı^−3537ȷ^v̂=\frac{12}{37}î-\frac{35}{37}ĵv^=3712ı^−3735ȷ^

v^=35ı^−12ȷ^v̂=35î-12ĵv^=35ı^−12ȷ^

v^=3537ı^−1237ȷ^v̂=\frac{35}{37}î-\frac{12}{37}ĵv^=3735ı^−3712ȷ^

Unit Vectors and i & j Notation Explained: Definition, Examples, Practice & Video Lessons

0

Concept

i & j Notation

Video duration:

6m

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.

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0

Problem

If vector $v ⃗ = 12 \hat{ı} - 2 \hat{ȷ}$ and vector $u⃗=5î+20ĵ$ calculate $2v ⃗-2u ⃗$ using $î$ and $ĵ$ notation.

A

$14î-44ĵ$

B

$7î-18ĵ$

C

$14î+44ĵ$

D

$14î-36ĵ$

0

Problem

If vector $v ⃗ =$ $11ĵ$ and vector $10î-25ĵ$ calculate $v⃗+$\frac$15u⃗$ using $î$ & $ĵ$ notation.

A

$13î+14ĵ$

B

$2î+14ĵ$

C

$13î-5ĵ$

D

$2î+6ĵ$

0

Problem

If vector $a ⃗ equals 20 dotless i hat$ and vector $b⃗=50ĵ$ calculate $a ⃗-b ⃗$ using $î$ and $ĵ$ notation.

A

$-30ĵ$

B

$20î-50ĵ$

C

$30î̂$

D

$20î+50ĵ$

0

Concept

Unit Vector in the Direction of a Given Vector

Video duration:

4m

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.

0

Example

Unit Vector in the Direction of a Given Vector Example 1

Video duration:

2m

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.

0

Problem

Find the unit vector in the direction of $a ⃗ = 6 \hat{ı} + 3 \hat{ȷ}$ .

A

$â=3$\surd$5ĵ$

B

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

C

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

D

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

0

Problem

Find the unit vector in the direction of $v ⃗ = 12 \hat{ı} - 35 \hat{ȷ}$ .

A

$v̂=$\frac{12}{37}$î-$\frac{35}{37}$ĵ$

B

$v̂=37î̂$

C

$v̂=35î-12ĵ$

D

$v̂=$\frac{35}{37}$î-$\frac{12}{37}$ĵ$

0

Example

Unit Vector in the Direction of a Given Vector Example 2

Video duration:

5m

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.

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

i & j Notation

i & j Notation Video Summary

If vector $v ⃗ = 12 \hat{ı} - 2 \hat{ȷ}$ and vector $u⃗=5î+20ĵ$ calculate $2v ⃗-2u ⃗$ using $î$ and $ĵ$ notation.

If vector $v ⃗ =$ $11ĵ$ and vector $10î-25ĵ$ calculate $v⃗+\(\frac\)15u⃗$ using $î$ & $ĵ$ notation.

If vector $a ⃗ equals 20 dotless i hat$ and vector $b⃗=50ĵ$ calculate $a ⃗-b ⃗$ using $î$ and $ĵ$ 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 $a ⃗ = 6 \hat{ı} + 3 \hat{ȷ}$ .

Find the unit vector in the direction of $v ⃗ = 12 \hat{ı} - 35 \hat{ȷ}$ .

Unit Vector in the Direction of a Given Vector Example 2

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

Your Algebra and Trigonometry tutors

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

i & j Notation

i & j Notation Video Summary

If vector v⃗=12ı^−2ȷ^v⃗=12î-2ĵ 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 ⃗=11ȷ^11ĵ11ȷ^ and vector 10ı^−25ȷ^10î-25ĵ10ı^−25ȷ^ calculate v⃗+15u⃗v⃗+\(\frac\)15u⃗v⃗+51u⃗ using ı^îı^ & ȷ^ĵȷ^ notation.

If vector a⃗=20ı^a⃗=20î 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+3ĵ.

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

Unit Vector in the Direction of a Given Vector Example 2

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

Your Algebra and Trigonometry tutors

If vector $v ⃗ = 12 \hat{ı} - 2 \hat{ȷ}$ and vector $u⃗=5î+20ĵ$ calculate $2v ⃗-2u ⃗$ using $î$ and $ĵ$ notation.

If vector $v ⃗ =$ $11ĵ$ and vector $10î-25ĵ$ calculate $v⃗+\(\frac\)15u⃗$ using $î$ & $ĵ$ notation.

If vector $a ⃗ equals 20 dotless i hat$ and vector $b⃗=50ĵ$ calculate $a ⃗-b ⃗$ using $î$ and $ĵ$ notation.

Find the unit vector in the direction of $a ⃗ = 6 \hat{ı} + 3 \hat{ȷ}$ .

Find the unit vector in the direction of $v ⃗ = 12 \hat{ı} - 35 \hat{ȷ}$ .