Vector A has magnitude 8.00 m and is in the x y -plane at an angle of 127^∘ counterclockwise fr

step 1 Here in this problem one vector has magnitude and direction given its angle from positive x -axi

Vector A has magnitude 8.00 m and is in the x y -plane at...

Vector $A$ has magnitude $8.00 \mathrm{~m}$ and is in the $x y$ -plane at an angle of $127^{\circ}$ counterclockwise from the $+x$ -axis $\left(37^{\circ}\right.$ past the $+y$ -axis). What are the magnitude and direction of vector $\vec{B}$ if the $\operatorname{sum} \vec{A}+\vec{B}$ is in the $-y$ -direction and has magnitude $12.0 \mathrm{~m} ?$

Key Concepts

Vector Decomposition

This concept involves breaking a vector into its horizontal and vertical components based on its magnitude and direction. It is essential for converting a vector from its polar form (magnitude and angle) to Cartesian coordinates (x and y components), which simplifies the process of performing vector addition and other operations.

Vector Addition

Vector addition is the process of finding the resultant vector when two or more vectors are combined. This is typically done by adding the corresponding components of the vectors. Understanding this concept allows one to determine unknown vectors when the resultant or other contributing vectors are given.

Trigonometric Analysis

Trigonometric analysis uses sine, cosine, and sometimes tangent functions to determine the component values of vectors from their magnitude and direction. This approach is crucial for accurately resolving vectors in problems where the vector's direction is given by an angle with respect to an axis.

Resultant Vectors and Directional Constraints

When a problem specifies that the resultant vector has a given magnitude and direction, it provides constraints that can be used to solve for unknown vectors. This requires setting up equations based on the predetermined direction (for instance, noting that if a vector is along the y-axis, its x component must be zero) and then using vector addition principles to satisfy these conditions.

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