Vector 𝐀 has a magnitude of 63 units and points due west, while vector 𝐁 has the same magnitude

step 1 All right, welcome back. We are adding up some more vectors today. So we have a vector that poin

Vector 𝐀 has a magnitude of 63 units and points due west,...

Vector $\overrightarrow{\mathbf{A}}$ has a magnitude of 63 units and points due west, while vector $\overrightarrow{\mathbf{B}}$ has the same magnitude and points due south. Find the magnitude and direction of (a) $\overrightarrow{\mathbf{A}}+\overrightarrow{\mathbf{B}}$ and (b) $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}} .$ Specify the directions relative to due west.

Key Concepts

Vector Representation and Components

Vectors are quantities that have both magnitude and direction, and they can be represented graphically or in coordinate form. This representation often involves breaking a vector into its horizontal and vertical components using unit vectors, which simplifies operations like addition and subtraction. By considering the components, one can apply algebraic methods to solve problem scenarios involving vectors.

Vector Addition and Subtraction

Vector addition and subtraction involves combining the corresponding components of each vector. When vectors are added, their resultant is obtained by summing their individual horizontal and vertical parts. Subtraction of vectors involves adding the negative of the vector being subtracted, which essentially reverses its direction prior to addition. This concept is critical as it provides a method to compute net effects from different vector quantities.

Magnitude Calculation

The magnitude, or length, of a vector is a measure of its size, irrespective of direction. After determining the components of a vector or a resultant vector from addition/subtraction, the magnitude can be calculated using the Pythagorean theorem by taking the square root of the sum of the squared components.

Direction of a Vector

The direction of a vector describes its orientation relative to a chosen reference line, often measured as an angle. This direction can be computed using inverse trigonometric functions, such as the arctan function, applied to the ratio of the vector's components. Specifying the angle relative to a specific axis or cardinal direction is essential for accurately describing the vector's orientation.

Transcript

00:01 All right, welcome back.

00:02 We are adding up some more vectors today.

00:06 So we have a vector that points due west, which i'll draw my compass rows right here, north, west, south, and east.

00:23 All right, so we have a vector point in due west of 63 units and another one.

00:31 We're adding the vector so the head of this vector right here is going to connect to the tail of the vector we're adding it to, and that vector goes due south, and it has a length of 63 units.

00:48 We'll call this vector a, this one right here, vector b.

00:54 All right, so a plus b is going to equal to, we'll call this vector c right here.

01:04 We've got green.

01:06 Green's a lovely color.

01:08 All right, vector c is this one right here.

01:14 All right, and so if we want to find the magnitude of vector c, we can use the pythagrin theorem.

01:24 That'll give us c is equal to a squared plus b squared.

01:30 So 63 squared plus 63 squared.

01:36 I can rewrite that if i want to as 2 times 63 squared, right? and that will be equal to about 89 units.

01:52 All right.

01:53 So that is the magnitude, the direction.

01:57 We can see that this is going to be an isosceles triangle, right? this side right here is equal to this side right here...

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