Solving for Vector Angles Given Vector Products and Lengths

When dealing with vector equations and their products, it’s essential to understand the relationship between the angle between two vectors and their dot products. This article delves into finding the angle between vectors A and B using the given vector equation and its length constraints.

Introduction to Vector Products and Angles

In vector algebra, the dot product (or scalar product) of two vectors is defined as:

( mathbf{a} cdot mathbf{b} ab cos theta )

where ( theta ) is the angle between the vectors ( mathbf{a} ) and ( mathbf{b} ), and ( a ) and ( b ) are the magnitudes of vectors ( mathbf{a} ) and ( mathbf{b} ), respectively.

Given Vector Equation and Lengths

We are given the vector equation:

Squaring this vector equation, we get:

( a^2 b^2 - 2ab cos theta c^2 1 )

Since ( c 1 ), we have:

( a^2 b^2 - 2ab cos theta 1 )

Solving for ( cos theta )

We know that ( mathbf{a} cdot mathbf{b} 1 ). Using the relationship between the lengths of the vectors and their dot product, we can solve for the angle ( theta ) as follows:

( cos theta -1 )

Thus, the angle ( theta ) between the vectors ( mathbf{a} ) and ( mathbf{b} ) is:

( theta pi )

Given that ( mathbf{a} cdot mathbf{b} 1 ) and substituting the values, we find:

( ab 1 )

For example:

( mathbf{a} 4hat{i} )

( 4b 1 Rightarrow b frac{1}{4} )

( mathbf{b}_1 -3hat{i} )

( mathbf{b}_2 -5hat{i} )

General Form and Solving for ( cos theta )

From the vector equation:

( vec{A} vec{B} vec{C} )

we can derive the squared magnitude of the vector ( vec{C} ) as:

( C^2 vec{C} cdot vec{C} vec{A} vec{B} cdot vec{A} vec{B} A^2 2AB cos theta B^2 )

Since ( AB C_1 ) then:

( A^2 2AB cos theta B^2 C_1^2 )

Solving both equations for ( cos theta ) to get:

( cos theta frac{C^2 - C_1^2}{2AB} )

This formula provides a systematic way to find the angle between two vectors given their dot product and lengths.

Conclusion

Understanding and utilizing the relationships between vector dot products, lengths, and angles is crucial in solving vector equations. By breaking down the problem step by step, we can find the angle between vectors using basic vector operations and properties.