Draw the vector C⃗ =1.5A⃗ −3B⃗ . The length and orientation of the vector will be graded. The location of the vector is not impo

I created the illustration found in the accompanying file.

There are two images included.

The upper one illustrates the impacts of:

- scaling vector A by a factor of 1.5, depicted in red with a dashed line.

- scaling vector B by -3, shown in purple with a dashed line.

The lower image displays the resultant vector: C = 1.5A - 3B.

The approach involves relocating the tail of vector -3B to the tip of vector 1.5A while maintaining the angles.

Next, an arrow is drawn from the tail of 1.5A to the position of -3B after this shift.

The arrow representing the result is vector C, marked with a black dashed line.
 

Additional details

A vector is a dimensional quantity characterized by both magnitude and direction.

Vectors can be represented by directed segments.

Here, the magnitude of a vector is expressed as | a |.

Vectors can be outlined as coordinate pairs indicating their locations within the Cartesian coordinate system:  a (a₁, a₂).

with length

Reversing the direction of vector a yields the vector -a, which retains the same magnitude yet points in the opposite direction.

Vector operations encompass adding and subtracting. The addition of vector a and vector b can be accomplished using a triangular method, aligning the start point of vector b with the end of vector a.

The result is achieved by drawing a segment from the beginning of vector a to the end of vector b, which creates a new vector c.

Thus, we have a + b = c.

When you add vector a to the negative of vector b (-b), the equation becomes a + (-b) = a - b.

Multiplying a vector by a scalar (represented as k) transforms it to k | a |.

If k > 0, the new vector points in the same direction as vector a, whereas if k < 0, it goes in the opposite direction.

The problem provides a partial diagram (see attached), and the total vector to find is C = 1.5A - 3B.

Scaling vector A by 1.5 results in a length that is 1.5 times that of vector A and maintains the same direction. Likewise, vector B, affected by -3, has a length that is three times that of vector B but points in the opposite direction. Hence, we derive vector C (refer to attached image) by drawing a line from the starting point of A to the endpoint of B.

Explore further

velocity position

understanding vector components

coherence in vector operations

Keywords: vector, addition, subtraction