Let \(\vec{a}=\hat{j}-\hat{k}\) and \(\vec{c}=\hat{i}-\hat{j}-\hat{k}\). The vector \(\vec{b}\) satisfying \(\vec{a} \times \vec{b}+\vec{c}=\overrightarrow{0}\) and \(\vec{a} \cdot \vec{b}=3\) is (1) \(2 \hat{i}-\hat{j}+2 \hat{k}\) (2) \(\hat{i}-\hat{j}-2 \hat{k}\) (3) \(\hat{i}+\hat{j}-2 \hat{k}\) (4) \(-\hat{i}+\hat{j}-2 \hat{k}\)

The vector cross product is a way to multiply two vectors in three-dimensional space. It results in another vector that is perpendicular to both of the original vectors. This is useful in physics when we want to find a direction that is orthogonal to a plane formed by two vectors. The symbol for the cross product is \(\times\). To compute the cross product of \( \vec{a} \) and \( \vec{b} \) given their components, we use the following determinant formula:
\[ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ a_x & a_y & a_z \ b_x & b_y & b_z \end{vmatrix} \]
This results in a new vector with its own components along \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\). For example, multiplying \( \vec{a} = \hat{j} - \hat{k} \) and \( \vec{b} = b_x \hat{i} + b_y \hat{j} + b_z \hat{k} \), we have:

\(\vec{a} \times \vec{b} = (\hat{j} - \hat{k}) \times (b_x \hat{i} + b_y \hat{j} + b_z \hat{k}) = -b_x \hat{j} - b_x \hat{k} + b_z \hat{i} \)

Note that the resulting vector components are orthogonal to both \( \vec{a} \) and \( \vec{b} \).

The dot product (or scalar product) is another way to multiply vectors, but unlike the cross product, it results in a scalar, not another vector. The dot product represents the magnitude of one vector projected onto the other and is given by:
\(\vec{a} \cdot \vec{b} = a_x b_x + a_y b_y + a_z b_z \)
For example, to find the dot product of \( \vec{a} = \hat{j} - \hat{k} \) and \( \vec{b} = -\hat{i} + \hat{j} - 2 \hat{k} \), we perform the operation:

\(\vec{a} \cdot \vec{b} = (0 \cdot -(1)) + (1 \cdot 1) + (-1 \cdot -2) = 1 + 2 = 3 \)

The dot product is often used to determine the angle between two vectors or to check for orthogonality (if \( \vec{a} \cdot \vec{b} = 0 \), then the vectors are orthogonal).

Vectors in physics are often broken down into their components to simplify calculations. Components are projections of a vector along the coordinate axes. For vector \( \vec{b} = b_x \hat{i} + b_y \hat{j} + b_z \hat{k} \), where \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) are unit vectors along the x, y, and z axes respectively, \( b_x, b_y, \)and \( b_z \) represent the magnitudes along these axes.
For example, if \( \vec{b} = -\hat{i} + \hat{j} - 2 \hat{k} \), its components are:

\(\text{Along x-axis:} \ -1\)
\(\text{Along y-axis:} \ 1 \)
\(\text{Along z-axis:} \ -2 \)

Understanding these components makes vector operations like addition, subtraction, dot product, and cross product easier since they can be performed component-wise. When combining vectors, simply add or subtract their corresponding components.