In R3
, you are given the point P=(6,1,−10)
 and the vector v=[−16,−8,5]
. If the arrow PQ⟶
 represents v
, then what is OQ−→−
?

Answer: The vector  OQ−→−
 is given by 
.  (Enter your answer within square brackets.)



If u,v
 and w
 are vectors in R2
 or R3
 such that u+v=w
, then the head-to-tail rule for addition of vectors can be illustrated using a triangle as in the diagram below. The vertices  A,B,C
 could be O,P,Q
 where O
 is the origin (0,0,0)
.

We are given the point \( P = (6, 1, -10) \) and the vector \( \mathbf{v} = [-16, -8, 5] \). The arrow \( \overrightarrow{PQ} \) represents the vector \( \mathbf{v} \), which means:

\[
\overrightarrow{PQ} = \mathbf{v} = [-16, -8, 5]
\]

To find the vector \( \overrightarrow{OQ} \), we use the fact that \( Q \) can be determined by adding the components of vector \( \mathbf{v} \) to the coordinates of point \( P \).

So, if \( P = (6, 1, -10) \) and \( \mathbf{v} = [-16, -8, 5] \), then the coordinates of point \( Q \) will be:

\[
Q = P + \mathbf{v} = (6, 1, -10) + (-16, -8, 5)
\]

Let's calculate the coordinates of \( Q \):

\[
Q_x = 6 + (-16) = -10
\]
\[
Q_y = 1 + (-8) = -7
\]
\[
Q_z = -10 + 5 = -5
\]

Thus, \( Q = (-10, -7, -5) \).

The vector \( \overrightarrow{OQ} \) is simply the position vector of \( Q \) relative to the origin \( O(0, 0, 0) \), which is the same as the coordinates of \( Q \):

\[
\overrightarrow{OQ} = [-10, -7, -5]
\]

So, the vector \( \overrightarrow{OQ} \) is:

\[
\boxed{[-10, -7, -5]}
\]

Would you like more details on vector operations or any clarification?

Here are 5 related questions for further exploration:
1. How do you calculate the magnitude of a vector in \( \mathbb{R}^3 \)?
2. What is the dot product of two vectors in \( \mathbb{R}^3 \), and how is it used?
3. How do you find the cross product of two vectors in \( \mathbb{R}^3 \)?
4. How do you determine if two vectors are perpendicular in \( \mathbb{R}^3 \)?
5. Can you explain how to compute the projection of one vector onto another in \( \mathbb{R}^3 \)?

**Tip**: When adding vectors, always apply the operation component-wise: add corresponding \( x \), \( y \), and \( z \)-coordinates.

In R3, you are given the point P=(6,1,−10) and the vector v=[−16,−8,5]. If the arrow PQ⟶ represents v, then what is OQ−→−?

Explore how to find the vector OQ in R3 given the point P=(6,1,-10) and vector v=[-16,-8,5]. This problem illustrates vector addition and the head-to-tail rule, suitable for high school students in Grades 10-12.

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