All categories

3 years ago

Find a vector i normal to the plane 2x + y + 2z = 4

Mathematics

1 answer:

m_a_m_a [10]3 years ago

4 0

Answer: <em>i</em> =2i+1j+2k

Step-by-step explanation:

The parametric equation of a plane is Ax+By+Cz=D and by definition the normal vector of the plane is n=Ai+Bj+Ck, where i,j,k are the canonic vectors

Then, in this case a normal vector to the given plane is i=2i+1j+2k

To prove that a vector defined as n=Ai+Bj+Ck is normal to the plane, let V1 and V2 be two vectors that are in the plane

If V1 ∈ Ax+By+Cz=D then Aa1+Bb1+Cc1=D

If V2 ∈ Ax+By+Cz=D then Aa2+Bb2+Cc2=D

The vector V1V2=V2-V1=(a2-a1)i+(b2-b1)j+(c2-c1)k

Two vectors are normal or perpendicular if the dot product is zero.

$V2V1_{o}n=0$

$(a2-a1)i+(b2-b1)j+(c2-c1)k_{o}Ai+Bj+Cj=0\\(a2-a1)A+(b2-b1)B+(c2-c1)C=0\\Aa2+Bb2+Cc2-(Aa1+Bb1+Cc1)=0\\D-D=0$

So, n is a normal vector to the plane.

You might be interested in

The graph of a linear relationship is shown. Which equation best represents the relationship between x and y in the graph

Finger [1]

Answer:

B. y = -3/4x + 5/2

Step-by-step explanation:

Take two points from the graph, let's use (-2, 4) and (2, 1):

Find slope:

m = (y₂ - y₁) / (x₂ - x₁)

= (1 - 4) / (2 - (-2))

= -3/4

Find y-intercept using slope above and anyone point:

y = mx + b

(1) = -3/4(2) + b

1 = -3/2 + b

b = 1 + 3/2

b = 5/2

Equation of line using m and b above:

y = mx + b

y = -3/4x + 5/2

8 0

3 years ago

To go to school in the morning. I first walk to the garden shed at 6 km/h to collect my bicycle. I then cycle to school at 15 km

HACTEHA [7]

Answer:

Step-by-step explanation:

3 0

3 years ago

Read 2 more answers

Hira is rollerblading at 9 m/s. Danni is climbing at 7 m/s. Amelia is cartwheeling at 2 m/s.

anyanavicka [17]

Answer:

Hira

Step-by-step explanation:

They are going at the fastest speed as 9 is the most

5 0

3 years ago

Read 2 more answers

3(4a + 2) = 2(a - 2)?

Likurg_2 [28]

Answer:

A= -1

Step-by-step explanation:

8 0

3 years ago

Need math help (last one wouldn't let me upload pic)

Scrat [10]

Answer:

$\text{Area of the figure}=54\text{ unit}^2$

Step-by-step explanation:

Please find the attachment.

Let us divide our given image in several parts and then we will find area of different parts.

First of all let us find the area of our red rectangle with side lengths 6 units and 7 units.

$\text{Area of rectangle}=\text{Length*Width}$

$\text{Area of red rectangle part}=6\text{ units}\times 7\text{ units}$

$\text{Area of red rectangle part}=42\text{ units}^2$

Now let us find area of yellow triangle on the top of red rectangle. We can see that base of triangle is 6 units and height is 1 unit.

$\text{Area of triangle}=\frac{1}{2}\times \text{Base*Height}$

$\text{Area of triangle}=\frac{1}{2}\times \text{6 units *1 unit}$

$\text{Area of triangle}=3\text{ unit}^2$

Let us find the area of green triangle.

$\text{Area of green triangle}=\frac{1}{2}\times \text{3 units *2 units}$

$\text{Area of green triangle}=3\text{ unit}^2$

Now we will find the area of blue triangle, whose base is 6 units and height is 2 units.

$\text{Area of blue triangle}=\frac{1}{2}\times \text{6 units *2 units}$

$\text{Area of blue triangle}=6\text{ units}^2$

Let us add all these areas to find the area of our figure.

$\text{Area of the figure}=42\text{ units}^2+3\text{ unit}^2+3\text{ unit}^2+6\text{ unit}^2$

$\text{Area of the figure}=(42+3+3+6)\text{ unit}^2$

$\text{Area of the figure}=54\text{ unit}^2$

Therefore, area of our given figure is 54 square units.

3 0

3 years ago