Projection of $\overrightarrow{c}$ on $\overrightarrow{a}$: Example

In summary, the projection of a vector on another vector is the component of the first vector that lies in the direction of the second vector, scaled by the length of the second vector. It can be calculated using a formula involving the dot product, and can be negative if the angle between the two vectors is greater than 90 degrees. The significance of this projection lies in its ability to find the component of the first vector that is parallel to the second vector, and it cannot be larger than the original vector.
  • #1
mathmari
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Hey! :eek:

To find the projection of $\overrightarrow{c}$ on $\overrightarrow{a}$ do we have to use the formula $$\frac{\overrightarrow{c} \cdot \overrightarrow{a}}{||\overrightarrow{a}||^2}\overrightarrow{a}$$ ?? (Wondering)

For example, if we have $\overrightarrow{c} =(4, 2, -6)$ and $\overrightarrow{a}=(-2, 2, 2)$ :

$$\frac{(4, 2, -6) \cdot (-2, 2, 2)}{||(-2, 2, 2)||^2}(-2, 2, 2)=\frac{-8+4-12}{4+4+4}(-2, 2, 2)=\frac{-16}{12}(-2, 2, 2)=\frac{-4}{3}(-2, 2, 2)$$

Is this the asked projection?? (Wondering)
 
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  • #2
Yes, that is correct, assuming you are to compute the vector projection. (Yes)
 
  • #3
Rido12 said:
Yes, that is correct, assuming you are to compute the vector projection. (Yes)

Great! Thank you! (Yes)
 

FAQ: Projection of $\overrightarrow{c}$ on $\overrightarrow{a}$: Example

What is the definition of projection of a vector on another vector?

The projection of a vector $\overrightarrow{c}$ on another vector $\overrightarrow{a}$ is the component of $\overrightarrow{c}$ that lies in the direction of $\overrightarrow{a}$, scaled by the length of $\overrightarrow{a}$. It can also be described as the shadow of $\overrightarrow{c}$ along the direction of $\overrightarrow{a}$.

How is the projection of a vector on another vector calculated?

The projection of a vector $\overrightarrow{c}$ on another vector $\overrightarrow{a}$ can be calculated using the formula:

$\text{proj}_{\overrightarrow{a}} \overrightarrow{c} = \frac{\overrightarrow{c} \cdot \overrightarrow{a}}{\overrightarrow{a} \cdot \overrightarrow{a}} \cdot \overrightarrow{a}$

Where $\cdot$ represents the dot product.

Can the projection of a vector on another vector be negative?

Yes, the projection of a vector $\overrightarrow{c}$ on another vector $\overrightarrow{a}$ can be negative if the angle between the two vectors is greater than 90 degrees. This means that the component of $\overrightarrow{c}$ along the direction of $\overrightarrow{a}$ is in the opposite direction of $\overrightarrow{a}$.

What is the significance of the projection of a vector on another vector?

The projection of a vector $\overrightarrow{c}$ on another vector $\overrightarrow{a}$ can be used to find the component of $\overrightarrow{c}$ that is parallel to $\overrightarrow{a}$. This is useful in many physical and mathematical applications, such as finding the work done by a force in a specific direction.

Can the projection of a vector on another vector be larger than the original vector?

No, the projection of a vector $\overrightarrow{c}$ on another vector $\overrightarrow{a}$ cannot be larger than the original vector $\overrightarrow{c}$. The length of the projection is always less than or equal to the length of the original vector, and it is only equal when the two vectors are parallel.

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