How do you find vector parametric equation for the line through the point P=(−4,−5,3) perpendicular to the plane 3x−4y+3z=−1?

Answer 1

#vecr=(-4,-5,3)+t(3,-4,3)#

The vector normal to the plane is #vecn=(3,-4,3)# The point P is #vecr_o=(-4,-5,3)# Equation is #vecr=r_o +tvecn# So the equation of the line is #vecr=(-4,-5,3)+t(3,-4,3)#
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Answer 2

To find the vector parametric equation for the line through the point ( P = (-4, -5, 3) ) perpendicular to the plane ( 3x - 4y + 3z = -1 ), follow these steps:

  1. Find the normal vector of the given plane. In this case, the coefficients of ( x ), ( y ), and ( z ) in the equation of the plane represent the components of the normal vector. So, the normal vector of the plane is ( \vec{n} = \langle 3, -4, 3 \rangle ).

  2. Let ( \vec{P_0} ) be the position vector of point ( P ), which is ( \vec{P_0} = \langle -4, -5, 3 \rangle ).

  3. Since the line is perpendicular to the plane, the direction of the line must be parallel to the normal vector of the plane. Thus, the direction vector of the line ( \vec{v} ) is the same as the normal vector of the plane, which is ( \vec{v} = \langle 3, -4, 3 \rangle ).

  4. Now, we can write the vector parametric equation of the line using the point ( P ) and the direction vector ( \vec{v} ). It is given by: [ \vec{r}(t) = \vec{P_0} + t\vec{v} ] where ( t ) is a parameter representing any real number.

  5. Substituting the values of ( \vec{P_0} ) and ( \vec{v} ): [ \vec{r}(t) = \langle -4, -5, 3 \rangle + t\langle 3, -4, 3 \rangle ]

  6. Simplify to get the vector parametric equation of the line: [ \vec{r}(t) = \langle -4 + 3t, -5 - 4t, 3 + 3t \rangle ]

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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