91Ó°ÊÓ

Vector addition is a fundamental operation in vector calculus. It involves combining two or more vectors to get a resultant vector. The concept relies on adding the corresponding components of each vector.
For instance, given two vectors \( \mathbf{u} = (u_1, u_2, u_3) \) and \( \mathbf{v} = (v_1, v_2, v_3) \), the sum \( \mathbf{u} + \mathbf{v} \) is calculated as follows:
\[ \mathbf{u} + \mathbf{v} = (u_1 + v_1, u_2 + v_2, u_3 + v_3) \]
This means you simply add the x-components, the y-components, and the z-components together.
For example, if we add \( \mathbf{u} = i + j \) and \( \mathbf{v} = -j - 2k \) (or in components as \( \mathbf{u} = (1, 1, 0) \) and \( \mathbf{v} = (0, -1, -2) \)), the result will be:
\[ \mathbf{u} + \mathbf{v} = (1+0, 1-1, 0-2) = (1, 0, -2) \]
This resultant vector combines the influences of both vectors in a spatial sense, providing a new direction and magnitude.

Vector subtraction involves finding the difference between two vectors, which is achieved by subtracting each corresponding component.
Given vectors \( \mathbf{u} = (u_1, u_2, u_3) \) and \( \mathbf{v} = (v_1, v_2, v_3) \), the operation \( \mathbf{u} - \mathbf{v} \) is expressed as:
\[ \mathbf{u} - \mathbf{v} = (u_1 - v_1, u_2 - v_2, u_3 - v_3) \]
This means subtract the x-component of \( \mathbf{v} \) from that of \( \mathbf{u} \), the y-component from \( \mathbf{u} \), and similarly with the z-components.
For our specific vectors \( \mathbf{u} = i + j \) and \( \mathbf{v} = -j - 2k \), set in component form as \( \mathbf{u} = (1, 1, 0) \) and \( \mathbf{v} = (0, -1, -2) \), their difference computes to:
\[ \mathbf{u} - \mathbf{v} = (1-0, 1+1, 0+2) = (1, 2, 2) \]
Vector subtraction results in a new vector representing the relative direction between the two original vectors. It essentially shows the change needed to get from \( \mathbf{v} \) to \( \mathbf{u} \).

Scalar multiplication in vector calculus involves scaling a vector by a real number, called the scalar. The scaling affects each component of the vector equally, either stretching or compressing it.
Given a vector \( \mathbf{u} = (u_1, u_2, u_3) \) and a scalar \( k \), the product \( k\mathbf{u} \) is calculated by:
\[ k \mathbf{u} = (k u_1, k u_2, k u_3) \]
This operation applies the scalar \( k \) to each component of the vector.
For example, if \( 3\mathbf{u} \) where \( \mathbf{u} = i + j \), translates to multiplying each component by 3:
\[ 3 \mathbf{u} = 3(1, 1, 0) = (3, 3, 0) \]
Similarly, \( \frac{1}{2} \mathbf{v} \) for \( \mathbf{v} = -j - 2k \) means multiplying each component by \( \frac{1}{2} \):
\[ \frac{1}{2} \mathbf{v} = \frac{1}{2}(0, -1, -2) = (0, -\frac{1}{2}, -1) \]
Combining these two, \( 3 \mathbf{u} - \frac{1}{2} \mathbf{v} \), requires calculating the difference:
\[ (3, 3, 0) - (0, -\frac{1}{2}, -1) = (3, 3 + \frac{1}{2}, 0 + 1) = (3, \frac{7}{2}, 1) \]
Scalar multiplication is a key operation for determining vector magnitude and direction, often utilized in vector projections and forces.