If a and b are two non collinear vectors then find the value of question attached herein ??

To find the value of something involving two non-collinear vectors \( \mathbf{a} \) and \( \mathbf{b} \), we first need to clarify a few concepts. Non-collinear vectors are those that do not lie along the same line; they have different directions. This property allows us to perform various operations with them, such as finding their cross product, which gives us a vector that is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \). Let's delve deeper into what you might be looking for regarding these two vectors.

Understanding Non-Collinearity

When we say that vectors \( \mathbf{a} \) and \( \mathbf{b} \) are non-collinear, it implies that they form a plane together. This property is essential in vector operations. For instance, if you visualize these vectors as arrows in space, they will create an angle between them, allowing for the computation of various vector properties.

Key Vector Operations

Here are a couple of significant operations you can perform with these vectors:

• Dot Product: The dot product of \( \mathbf{a} \) and \( \mathbf{b} \) is given by \( \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| |\mathbf{b}| \cos(\theta) \), where \( \theta \) is the angle between the two vectors. This operation yields a scalar value.
• Cross Product: The cross product \( \mathbf{a} \times \mathbf{b} \) produces a vector that is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \). The magnitude of this vector is \( |\mathbf{a}| |\mathbf{b}| \sin(\theta) \), which is also significant as it represents the area of the parallelogram formed by the two vectors.

Calculating the Value

If your question is about finding a specific value derived from vectors \( \mathbf{a} \) and \( \mathbf{b} \), you might be looking for one of the following:

• The angle \( \theta \) between the vectors, which can be calculated using the dot product.
• The area of the parallelogram formed by the vectors, which can be determined from the magnitude of the cross product.
• A specific component of one vector in the direction of the other, which can be found using projection formulas.

Example Calculation

Let’s say you want to find the area of the parallelogram formed by \( \mathbf{a} \) and \( \mathbf{b} \). If \( \mathbf{a} = (2, 3) \) and \( \mathbf{b} = (1, 4) \), we can compute the cross product as follows:

First, represent the vectors in three-dimensional space by adding a zero z-component:

\( \mathbf{a} = (2, 3, 0) \) and \( \mathbf{b} = (1, 4, 0) \).

Now, the cross product \( \mathbf{a} \times \mathbf{b} \) is calculated using the determinant of a matrix:

\[\mathbf{a} \times \mathbf{b} = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \\2 & 3 & 0 \\1 & 4 & 0\end{vmatrix}\]

This results in a vector whose z-component gives the area of the parallelogram:

Calculating that, we find the area is \( |6| = 6 \) square units.

In summary, to determine the specific value you're interested in regarding the non-collinear vectors \( \mathbf{a} \) and \( \mathbf{b} \), consider what operation or property you need. Whether it’s the angle between them, the area of the shape they form, or something else, understanding their relationship through vector operations opens up a world of possibilities in vector mathematics.

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