Question

Find the component form of v given its magnitude and the angle it makes with the positive x-axis. ‖v‖=14,v in the direction of 10i 6j

Answer 1

The component form of a vector given its magnitude and direction can be found by normalizing the direction vector and then multiplying its components by the vector's magnitude. Apply cosine and sine functions to find the components, with magnitude as the scalar.

Step-by-step explanation:

To find the component form of a vector given its magnitude and direction, you can use the cosine and sine functions. The direction of vector v is provided as a direction vector, 10i 6j, which we can use to determine the angle the vector makes with the positive x-axis.

The first step is to normalize the direction vector. The normalized direction vector (unit vector) will have components that correspond to the cosine and sine of the angle of the vector with respect to the x-axis. For the direction vector 10i + 6j, we calculate the magnitude of this vector as the square root of (10^2 + 6^2). Once we have the magnitude of the direction vector, we divide each component by this magnitude to get the unit vector.

After normalizing the direction vector, we can multiply its components by the magnitude of vector v (which is 14 in this case) to get the component form of vector v.

Here's a step-by-step calculation:

1. 
   
3. Calculate the magnitude of the direction vector: magnitude = √(10^2 + 6^2)
4. 
   
6. Normalize the direction vector: unitVector = (10 / magnitude)i + (6 / magnitude)j
7. 
   
9. Multiply the unit vector by the magnitude of vector v: componentForm = (14 * unitVector)