To solve this problem, let's start with the given information:

The vectors are: $\vec{D} = 5\mathbf{i} - 2\mathbf{j}, \quad \vec{B} = -7\mathbf{i} - 2\mathbf{j}, \quad \vec{A} = -3\mathbf{i} - 6\mathbf{j}$

We want to find constants $a$ and $b$ such that: $a\vec{D} + b\vec{B} + \vec{A} = \vec{0}$

Substituting the vectors into the equation: $a(5\mathbf{i} - 2\mathbf{j}) + b(-7\mathbf{i} - 2\mathbf{j}) + (-3\mathbf{i} - 6\mathbf{j}) = \vec{0}$

Expanding this: $(5a - 7b - 3)\mathbf{i} + (-2a - 2b - 6)\mathbf{j} = \vec{0}$

Since the left side must be equal to the zero vector, we can separate this into two equations:

1. $5a - 7b - 3 = 0$
2. $-2a - 2b - 6 = 0$

Let's solve this system of linear equations.

Starting with:

1. $5a - 7b = 3$
2. $-2a - 2b = 6$

Let's solve this system to find $a$ and $b$.The solution to the system of equations is: $a = -\frac{3}{2}, \quad b = -\frac{3}{2}$

Thus, the values of the constants are: $a = -1.5, \quad b = -1.5$

Would you like a more detailed explanation or have any further questions?

Related Questions:

1. How do we verify that $a$ and $b$ satisfy the original vector equation?
2. What is the geometrical interpretation of the result $a\vec{D} + b\vec{B} + \vec{A} = \vec{0}$?
3. How can this problem be extended to three-dimensional vectors?
4. How would the solution change if the vectors were expressed in terms of different basis vectors?
5. What other methods can we use to solve a system of linear equations involving vectors?

Tip:

When solving systems of equations involving vectors, always equate the corresponding components to create simpler scalar equations that can be solved using algebraic techniques.