For the vectors A⃗ and B⃗ in Fig. E 1.22, use the method of components to find the magnitude and

For the vectors A⃗ and B⃗ in Fig. E 1.22, use the method of...

For the vectors $\vec{A}$ and $\vec{B}$ in Fig. $\mathrm{E} 1.22,$ use the method of components to find the magnitude and direction of (a) the vector sum $\overrightarrow{\boldsymbol{A}}+\overrightarrow{\boldsymbol{B}} ;$ (b) the vector $\operatorname{sum} \overrightarrow{\boldsymbol{B}}+\overrightarrow{\boldsymbol{A}} ;$ (c) the vector difference $\overrightarrow{\boldsymbol{A}}-\overrightarrow{\boldsymbol{B}} ;$ (d) the vector difference $\vec{B}-\vec{A}$.

Key Concepts

Magnitude Calculation

Magnitude refers to the length or size of a vector, which can be determined using the Pythagorean theorem when the vector is expressed in its component form. The formula involves taking the square root of the sum of the squares of its individual components.

Vector Subtraction

Vector subtraction involves finding the difference between two vectors, which can be interpreted as adding a negative vector. Similar to vector addition, the subtraction is performed by individually subtracting the corresponding components, leading to the resultant vector difference.

Direction Calculation

The direction of a vector is determined by the angle it makes with a reference axis, often the horizontal axis, and is calculated using trigonometric functions such as the arctan of the ratio of the vertical to horizontal components. This angle provides the orientation of the vector in space.

Method of Components

This approach involves breaking vectors into their horizontal (x) and vertical (y) components and then combining these components separately to determine the resultant vector. It simplifies the process of adding or subtracting vectors, making calculations systematic and straightforward.

Vector Addition

Vector addition is the process of combining two or more vectors to produce a resultant vector. When vectors are added component-wise, their x- and y-components are summed individually, yielding the corresponding components of the overall vector sum.

Transcript

00:01 Hi, here in this given problem we have to focus only the two vectors, vector a and vector b.

00:06 That's why i am drawing only these two vectors over here.

00:12 First of all, this is x -axis, this is y -axis, vector b, a vector a that is along negative y -axis and this is vector b making an angle of 30 degree with a positive y -axis.

00:33 So, this is vector b whose length is 15 .0 meter and vector a having a magnitude 8 .00 meter.

00:50 This angle over here that is 30 .0 degree.

00:56 So, first of all, we have to find, we will find the components of vector b and vector a along x and y -axis.

01:08 So, here it will become by and here it will be bx.

01:14 Now ax that is 0 as vector a is purely along y -axis.

01:19 So, ay and this is along negative y -axis.

01:22 So, ay is equal to minus 8 .00 meter.

01:29 Bx that will be b sin 30 degree means 15 .0 into sin 30 that is half.

01:40 So, it comes out to be equal to 7 .5 meter and by that is b cos 30 degree means 15 .0 cos 30 degree.

01:54 So, it comes out to be equal to 13 .0 meter.

02:00 So, in the first part of the problem where we have to find vector a plus vector b.

02:06 So, first of all we find x component of this addition of a and b and that will be given by ax plus bx.

02:16 So, 0 plus 7 .5 means this is 7 .5 meter only and then we find its y component that will be ay plus by means this is minus 8 plus 13.

02:41 So, it comes out to be equal to plus 5 .0 meter.

02:46 So, magnitude of a plus b that will be given by using vector sum relation square of x component 7 .5 square plus y component 5 square and it is calculated to be equal to 9 .01 meter...

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