If P ⃗ + Q ⃗ + R ⃗ = 0 \vec{p} + \vec{q} + \vec{r} = 0 P + Q + R = 0 , ∣ P ⃗ ∣ = 6 |\vec{p}| = 6 ∣ P ∣ = 6 , ∣ Q ⃗ ∣ = 10 |\vec{q}| = 10 ∣ Q ∣ = 10 , And P ⃗ ⋅ Q ⃗ = 30 \vec{p} \cdot \vec{q} = 30 P ⋅ Q = 30 , Then Find ∣ R ⃗ ∣ |\vec{r}| ∣ R ∣ .

Mar 12, 2025 by ADMIN 257 views

$If $\vec{p} + \vec{q} + \vec{r} = 0$, $|\vec{p}| = 6$, $|\vec{q}| = 10$, and $\vec{p} \cdot \vec{q} = 30$, then find $|\vec{r}|$$

Introduction

In this article, we will explore the concept of vectors and their properties. We will use the given information to find the magnitude of vector $\vec{r}$ , which is a crucial concept in mathematics and physics. The given information includes the sum of three vectors, the magnitudes of two vectors, and the dot product of two vectors. We will use these properties to find the magnitude of vector $\vec{r}$ .

Properties of Vectors

Vectors are mathematical objects that have both magnitude and direction. They can be represented graphically as arrows in a coordinate system. The magnitude of a vector is its length, while the direction is the angle it makes with the positive x-axis. The dot product of two vectors is a scalar value that represents the amount of "similarity" between the two vectors.

The Given Information

We are given that $\vec{p} + \vec{q} + \vec{r} = 0$ . This means that the sum of the three vectors is equal to the zero vector. We are also given that $|\vec{p}| = 6$ and $|\vec{q}| = 10$ . This means that the magnitude of vector $\vec{p}$ is 6 units, and the magnitude of vector $\vec{q}$ is 10 units. Finally, we are given that $\vec{p} \cdot \vec{q} = 30$ . This means that the dot product of vectors $\vec{p}$ and $\vec{q}$ is 30.

Using the Properties of Vectors to Find $|\vec{r}|$

We can use the properties of vectors to find the magnitude of vector $\vec{r}$ . Since $\vec{p} + \vec{q} + \vec{r} = 0$ , we can write:

\vec{r} = -(\vec{p} + \vec{q})

We can use the property of the dot product to find the magnitude of vector $\vec{r}$ . The dot product of a vector with itself is equal to the square of its magnitude. Therefore, we can write:

|\vec{r}|^2 = \vec{r} \cdot \vec{r} = (-\vec{p} - \vec{q}) \cdot (-\vec{p} - \vec{q})

Expanding the dot product, we get:

|\vec{r}|^2 = \vec{p} \cdot \vec{p} + 2\vec{p} \cdot \vec{q} + \vec{q} \cdot \vec{q}

We know that $\vec{p} \cdot \vec{p} = |\vec{p}|^2 = 6^2 = 36$ and $\vec{q} \cdot \vec{q} = |\vec{q}|^2 = 10^2 = 100$ . We are also given that $\vec{p} \cdot \vec{q} = 30$ . Therefore, we can substitute these values into the equation:

|\vec{r}|^2 = 36 + 2(30) + 100

Simplifying the equation, we get:

|\vec{r}|^2 = 36 + 60 + 100

|\vec{r}|^2 = 196

Taking the square root of both sides, we get:

|\vec{r}| = \sqrt{196}

|\vec{r}| = 14

Therefore, the magnitude of vector $\vec{r}$ is 14 units.

Conclusion

In this article, we used the properties of vectors to find the magnitude of vector $\vec{r}$ . We were given that $\vec{p} + \vec{q} + \vec{r} = 0$ , $|\vec{p}| = 6$ , $|\vec{q}| = 10$ , and $\vec{p} \cdot \vec{q} = 30$ . We used the dot product and the properties of vectors to find the magnitude of vector $\vec{r}$ . The magnitude of vector $\vec{r}$ is 14 units.

References

[1] "Vector Algebra" by Wikipedia
[2] "Vector Calculus" by MIT OpenCourseWare
[3] "Linear Algebra and Its Applications" by Gilbert Strang

Q&A: If $\vec{p} + \vec{q} + \vec{r} = 0$ , $|\vec{p}| = 6$ , $|\vec{q}| = 10$ , and $\vec{p} \cdot \vec{q} = 30$ , then find $|\vec{r}|$

Q: What is the relationship between the vectors $\vec{p}$ , $\vec{q}$ , and $\vec{r}$ ?

A: The vectors $\vec{p}$ , $\vec{q}$ , and $\vec{r}$ are related by the equation $\vec{p} + \vec{q} + \vec{r} = 0$ . This means that the sum of the three vectors is equal to the zero vector.

Q: What is the magnitude of vector $\vec{p}$ ?

A: The magnitude of vector $\vec{p}$ is 6 units.

Q: What is the magnitude of vector $\vec{q}$ ?

A: The magnitude of vector $\vec{q}$ is 10 units.

Q: What is the dot product of vectors $\vec{p}$ and $\vec{q}$ ?

A: The dot product of vectors $\vec{p}$ and $\vec{q}$ is 30.

Q: How can we use the properties of vectors to find the magnitude of vector $\vec{r}$ ?

A: We can use the properties of vectors to find the magnitude of vector $\vec{r}$ by using the equation $\vec{r} = -(\vec{p} + \vec{q})$ . We can then use the dot product to find the magnitude of vector $\vec{r}$ .

Q: What is the magnitude of vector $\vec{r}$ ?

A: The magnitude of vector $\vec{r}$ is 14 units.

Q: Can we use the same method to find the magnitude of vector $\vec{r}$ if the dot product of vectors $\vec{p}$ and $\vec{q}$ is not given?

A: No, we cannot use the same method to find the magnitude of vector $\vec{r}$ if the dot product of vectors $\vec{p}$ and $\vec{q}$ is not given. The dot product is a crucial piece of information that is needed to find the magnitude of vector $\vec{r}$ .

Q: Can we use the same method to find the magnitude of vector $\vec{r}$ if the magnitudes of vectors $\vec{p}$ and $\vec{q}$ are not given?

A: No, we cannot use the same method to find the magnitude of vector $\vec{r}$ if the magnitudes of vectors $\vec{p}$ and $\vec{q}$ are not given. The magnitudes of vectors $\vec{p}$ and $\vec{q}$ are crucial pieces of information that are needed to find the magnitude of vector $\vec{r}$ .

Q: Can we use the same method to find the magnitude of vector $\vec{r}$ if the vectors $\vec{p}$ , $\vec{q}$ , and $\vec{r}$ are not related by the equation $\vec{p} + \vec{q} + \vec{r} = 0$ ?

A: No, we cannot use the same method to find the magnitude of vector $\vec{r}$ if the vectors $\vec{p}$ , $\vec{q}$ , and $\vec{r}$ are not related by the equation $\vec{p} + \vec{q} + \vec{r} = 0$ . The equation $\vec{p} + \vec{q} + \vec{r} = 0$ is a crucial piece of information that is needed to find the magnitude of vector $\vec{r}$ .

Conclusion

References

[1] "Vector Algebra" by Wikipedia
[2] "Vector Calculus" by MIT OpenCourseWare
[3] "Linear Algebra and Its Applications" by Gilbert Strang

If P ⃗ + Q ⃗ + R ⃗ = 0 \vec{p} + \vec{q} + \vec{r} = 0 P + Q + R = 0 , ∣ P ⃗ ∣ = 6 |\vec{p}| = 6 ∣ P ∣ = 6 , ∣ Q ⃗ ∣ = 10 |\vec{q}| = 10 ∣ Q ∣ = 10 , And P ⃗ ⋅ Q ⃗ = 30 \vec{p} \cdot \vec{q} = 30 P ⋅ Q = 30 , Then Find ∣ R ⃗ ∣ |\vec{r}| ∣ R ∣ .

Introduction

Properties of Vectors

The Given Information

Using the Properties of Vectors to Find $|\vec{r}|$

Conclusion

References

Further Reading

Q&A: If $\vec{p} + \vec{q} + \vec{r} = 0$ , $|\vec{p}| = 6$ , $|\vec{q}| = 10$ , and $\vec{p} \cdot \vec{q} = 30$ , then find $|\vec{r}|$

Q: What is the relationship between the vectors $\vec{p}$ , $\vec{q}$ , and $\vec{r}$ ?

Q: What is the magnitude of vector $\vec{p}$ ?

Q: What is the magnitude of vector $\vec{q}$ ?

Q: What is the dot product of vectors $\vec{p}$ and $\vec{q}$ ?

Q: How can we use the properties of vectors to find the magnitude of vector $\vec{r}$ ?

Q: What is the magnitude of vector $\vec{r}$ ?

Q: Can we use the same method to find the magnitude of vector $\vec{r}$ if the dot product of vectors $\vec{p}$ and $\vec{q}$ is not given?

Q: Can we use the same method to find the magnitude of vector $\vec{r}$ if the magnitudes of vectors $\vec{p}$ and $\vec{q}$ are not given?

Q: Can we use the same method to find the magnitude of vector $\vec{r}$ if the vectors $\vec{p}$ , $\vec{q}$ , and $\vec{r}$ are not related by the equation $\vec{p} + \vec{q} + \vec{r} = 0$ ?

Conclusion

References

Further Reading

Introduction

Properties of Vectors

The Given Information

Using the Properties of Vectors to Find ∣r⃗∣|\vec{r}|∣r∣

Conclusion

References

Further Reading

Q&A: If p⃗+q⃗+r⃗=0\vec{p} + \vec{q} + \vec{r} = 0p​+q​+r=0, ∣p⃗∣=6|\vec{p}| = 6∣p​∣=6, ∣q⃗∣=10|\vec{q}| = 10∣q​∣=10, and p⃗⋅q⃗=30\vec{p} \cdot \vec{q} = 30p​⋅q​=30, then find ∣r⃗∣|\vec{r}|∣r∣

Q: What is the relationship between the vectors p⃗\vec{p}p​, q⃗\vec{q}q​, and r⃗\vec{r}r?

Q: What is the magnitude of vector p⃗\vec{p}p​?

Q: What is the magnitude of vector q⃗\vec{q}q​?

Q: What is the dot product of vectors p⃗\vec{p}p​ and q⃗\vec{q}q​?

Q: How can we use the properties of vectors to find the magnitude of vector r⃗\vec{r}r?

Q: What is the magnitude of vector r⃗\vec{r}r?

Q: Can we use the same method to find the magnitude of vector r⃗\vec{r}r if the dot product of vectors p⃗\vec{p}p​ and q⃗\vec{q}q​ is not given?

Q: Can we use the same method to find the magnitude of vector r⃗\vec{r}r if the magnitudes of vectors p⃗\vec{p}p​ and q⃗\vec{q}q​ are not given?

Q: Can we use the same method to find the magnitude of vector r⃗\vec{r}r if the vectors p⃗\vec{p}p​, q⃗\vec{q}q​, and r⃗\vec{r}r are not related by the equation p⃗+q⃗+r⃗=0\vec{p} + \vec{q} + \vec{r} = 0p​+q​+r=0?

Conclusion

References

Further Reading

Using the Properties of Vectors to Find $|\vec{r}|$

Q&A: If $\vec{p} + \vec{q} + \vec{r} = 0$ , $|\vec{p}| = 6$ , $|\vec{q}| = 10$ , and $\vec{p} \cdot \vec{q} = 30$ , then find $|\vec{r}|$

Q: What is the relationship between the vectors $\vec{p}$ , $\vec{q}$ , and $\vec{r}$ ?

Q: What is the magnitude of vector $\vec{p}$ ?

Q: What is the magnitude of vector $\vec{q}$ ?

Q: What is the dot product of vectors $\vec{p}$ and $\vec{q}$ ?

Q: How can we use the properties of vectors to find the magnitude of vector $\vec{r}$ ?

Q: What is the magnitude of vector $\vec{r}$ ?

Q: Can we use the same method to find the magnitude of vector $\vec{r}$ if the dot product of vectors $\vec{p}$ and $\vec{q}$ is not given?

Q: Can we use the same method to find the magnitude of vector $\vec{r}$ if the magnitudes of vectors $\vec{p}$ and $\vec{q}$ are not given?

Q: Can we use the same method to find the magnitude of vector $\vec{r}$ if the vectors $\vec{p}$ , $\vec{q}$ , and $\vec{r}$ are not related by the equation $\vec{p} + \vec{q} + \vec{r} = 0$ ?