Express a Vector in Component Form - Pre-Calculus

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Question

Express the following vector in component form:

Answer

When separating a vector into its component form, we are essentially creating a right triangle with the vector being the hypotenuse.

Therefore, we can find each component using the cos (for the x component) and sin (for the y component) functions:

$v_{x} = 27cos(55^{\circ}) = 15.5$

$v_y = 27sin(55^{\circ}) = 22.1$

We can now represent these two components together using the denotations i (for the x component) and j (for the y component).

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Question

Find , then find its magnitude. and are both vectors.

Answer

In vector addition, you simply add each component of the vectors to each other.

x component: .

y component: .

z component: .

The new vector is

.

To find the magnitude we use the formula,

$|a+b|=\sqrt{0^2+3^2+4^2}$

$|a+b|=\sqrt{9+16}$

$|a+b|=\sqrt{25}=5$

Thus its magnitude is 5.

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Question

Find the component form of the vector with

initial point

and

terminal point .

Answer

To find the vector in component form given the initial and terminal points, simply subtract the initial point from the terminal point.

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Question

Find the component form of the vector with

initial point

and

terminal point

Answer

To find the vector in component form given the initial and terminal points, simply subtract the initial point from the terminal point.

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Question

A bird flies 15 mph up at an angle of 45 degrees to the horizontal. What is the bird's velocity in component form?

Answer

Write the formula to find both the x and y-components of a vector.

$x=Vcos\theta$

$y=Vsin\theta$

Substitute the value of velocity and theta into the equations.

$x=15cos(45) = \frac{15\sqrt2}{2}$

$y=15sin(45) = \frac{15\sqrt2}{2}$

The vector is: $\left \langle \frac{15\sqrt2}{2},\frac{15\sqrt2}{2} \right \rangle$

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Question

Write this vector in component form:

Answer

In order to find the horizontal component, set up an equation involving cosine with 7 as the hypotenuse, since the side in the implied triangle that represents the horizontal component is adjacent to the 22-degree angle:

$\small cos(22)=\frac{x}{7}$ First, find the cosine of 22, then multiply by 7

$\small 6.49 \approx x$

To find the vertical component, set up an equation involving sine, since the side in the implied triangle that represents the vertical component is opposite the 22-degree angle:

$\small sin(22)=\frac{y}{7}$ First, find the sine of 22, then multiply by 7

$\small 2.62 \approx x$

We are almost done, but we need to make a small adjustment. The picture indicates that the vector points up and to the left, so the horizontal component, 6.49, should be negative:

$\small \small <-6.49, 2.62>$

Compare your answer with the correct one above

Question

Express the following vector in component form:

Answer

When separating a vector into its component form, we are essentially creating a right triangle with the vector being the hypotenuse.

Therefore, we can find each component using the cos (for the x component) and sin (for the y component) functions:

$v_{x} = 27cos(55^{\circ}) = 15.5$

$v_y = 27sin(55^{\circ}) = 22.1$

We can now represent these two components together using the denotations i (for the x component) and j (for the y component).

Compare your answer with the correct one above

Question

Find , then find its magnitude. and are both vectors.

Answer

In vector addition, you simply add each component of the vectors to each other.

x component: .

y component: .

z component: .

The new vector is

.

To find the magnitude we use the formula,

$|a+b|=\sqrt{0^2+3^2+4^2}$

$|a+b|=\sqrt{9+16}$

$|a+b|=\sqrt{25}=5$

Thus its magnitude is 5.

Compare your answer with the correct one above

Question

Find the component form of the vector with

initial point

and

terminal point .

Answer

To find the vector in component form given the initial and terminal points, simply subtract the initial point from the terminal point.

Compare your answer with the correct one above

Question

Find the component form of the vector with

initial point

and

terminal point

Answer

To find the vector in component form given the initial and terminal points, simply subtract the initial point from the terminal point.

Compare your answer with the correct one above

Question

A bird flies 15 mph up at an angle of 45 degrees to the horizontal. What is the bird's velocity in component form?

Answer

Write the formula to find both the x and y-components of a vector.

$x=Vcos\theta$

$y=Vsin\theta$

Substitute the value of velocity and theta into the equations.

$x=15cos(45) = \frac{15\sqrt2}{2}$

$y=15sin(45) = \frac{15\sqrt2}{2}$

The vector is: $\left \langle \frac{15\sqrt2}{2},\frac{15\sqrt2}{2} \right \rangle$

Compare your answer with the correct one above

Question

Write this vector in component form:

Answer

In order to find the horizontal component, set up an equation involving cosine with 7 as the hypotenuse, since the side in the implied triangle that represents the horizontal component is adjacent to the 22-degree angle:

$\small cos(22)=\frac{x}{7}$ First, find the cosine of 22, then multiply by 7

$\small 6.49 \approx x$

To find the vertical component, set up an equation involving sine, since the side in the implied triangle that represents the vertical component is opposite the 22-degree angle:

$\small sin(22)=\frac{y}{7}$ First, find the sine of 22, then multiply by 7

$\small 2.62 \approx x$

We are almost done, but we need to make a small adjustment. The picture indicates that the vector points up and to the left, so the horizontal component, 6.49, should be negative:

$\small \small <-6.49, 2.62>$

Compare your answer with the correct one above

Question

Express the following vector in component form:

Answer

When separating a vector into its component form, we are essentially creating a right triangle with the vector being the hypotenuse.

Therefore, we can find each component using the cos (for the x component) and sin (for the y component) functions:

$v_{x} = 27cos(55^{\circ}) = 15.5$

$v_y = 27sin(55^{\circ}) = 22.1$

We can now represent these two components together using the denotations i (for the x component) and j (for the y component).

Compare your answer with the correct one above

Question

Find , then find its magnitude. and are both vectors.

Answer

In vector addition, you simply add each component of the vectors to each other.

x component: .

y component: .

z component: .

The new vector is

.

To find the magnitude we use the formula,

$|a+b|=\sqrt{0^2+3^2+4^2}$

$|a+b|=\sqrt{9+16}$

$|a+b|=\sqrt{25}=5$

Thus its magnitude is 5.

Compare your answer with the correct one above

Question

Find the component form of the vector with

initial point

and

terminal point .

Answer

To find the vector in component form given the initial and terminal points, simply subtract the initial point from the terminal point.

Compare your answer with the correct one above

Question

Find the component form of the vector with

initial point

and

terminal point

Answer

To find the vector in component form given the initial and terminal points, simply subtract the initial point from the terminal point.

Compare your answer with the correct one above

Question

A bird flies 15 mph up at an angle of 45 degrees to the horizontal. What is the bird's velocity in component form?

Answer

Write the formula to find both the x and y-components of a vector.

$x=Vcos\theta$

$y=Vsin\theta$

Substitute the value of velocity and theta into the equations.

$x=15cos(45) = \frac{15\sqrt2}{2}$

$y=15sin(45) = \frac{15\sqrt2}{2}$

The vector is: $\left \langle \frac{15\sqrt2}{2},\frac{15\sqrt2}{2} \right \rangle$

Compare your answer with the correct one above

Question

Write this vector in component form:

Answer

In order to find the horizontal component, set up an equation involving cosine with 7 as the hypotenuse, since the side in the implied triangle that represents the horizontal component is adjacent to the 22-degree angle:

$\small cos(22)=\frac{x}{7}$ First, find the cosine of 22, then multiply by 7

$\small 6.49 \approx x$

To find the vertical component, set up an equation involving sine, since the side in the implied triangle that represents the vertical component is opposite the 22-degree angle:

$\small sin(22)=\frac{y}{7}$ First, find the sine of 22, then multiply by 7

$\small 2.62 \approx x$

We are almost done, but we need to make a small adjustment. The picture indicates that the vector points up and to the left, so the horizontal component, 6.49, should be negative:

$\small \small <-6.49, 2.62>$

Compare your answer with the correct one above

Question

Express the following vector in component form:

Answer

When separating a vector into its component form, we are essentially creating a right triangle with the vector being the hypotenuse.

Therefore, we can find each component using the cos (for the x component) and sin (for the y component) functions:

$v_{x} = 27cos(55^{\circ}) = 15.5$

$v_y = 27sin(55^{\circ}) = 22.1$

We can now represent these two components together using the denotations i (for the x component) and j (for the y component).

Compare your answer with the correct one above

Question

Find , then find its magnitude. and are both vectors.

Answer

In vector addition, you simply add each component of the vectors to each other.

x component: .

y component: .

z component: .

The new vector is

.

To find the magnitude we use the formula,

$|a+b|=\sqrt{0^2+3^2+4^2}$

$|a+b|=\sqrt{9+16}$

$|a+b|=\sqrt{25}=5$

Thus its magnitude is 5.

Compare your answer with the correct one above

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