Question

The vector(s) which is/are coplanar with vectors \(\hat{i}+\hat{j}+2 \hat{k}\) and \(\hat{i}+2 \hat{j}+\hat{k}\), and perpendicular to the vector \(\hat{i}+\hat{j}+\hat{k}\) is/are (A) \(\hat{j}-\hat{k}\) (B) \(-\hat{i}+\hat{j}\) (C) \(\hat{i}-\hat{j}\) (D) \(-\hat{j}+\hat{k}\)

Short answer

None of the options are perpendicular to \((\hat{i}+\hat{j}+\hat{k})\), so none are coplanar with both of the other vectors.

Step-by-step solution

- Find the Normal Vector

To determine the vectors that are coplanar with the given vectors, first we find a normal vector to the plane that contains them by taking their cross product.

- Calculate the Cross Product

The cross product of \((\hat{i}+\hat{j}+2\hat{k})\) and \((\hat{i}+2\hat{j}+\hat{k})\) can be computed using the determinant method or by components. This vector will be perpendicular to the plane containing the original two vectors.

- Analyze the Perpendicular Vector

The vector(s) perpendicular to \((\hat{i}+\hat{j}+\hat{k})\) will have a dot product of zero with it. We check each option (A), (B), (C), (D) to see which is perpendicular to this vector.

- Determine the Coplanar and Perpendicular Vector(s)

By calculating the dot product of each option with \((\hat{i}+\hat{j}+\hat{k})\), we find the vector(s) that satisfy the conditions of being coplanar with the given vectors and perpendicular to \((\hat{i}+\hat{j}+\hat{k})\).

- Select the Correct Answer

After calculating, we'll select the option(s) where the dot product with \((\hat{i}+\hat{j}+\hat{k})\) equals to zero, as those will be the correct answers.

Key concepts

Vector Cross Product

Understanding the vector cross product is crucial for solving problems involving three-dimensional vectors, such as finding a vector that is perpendicular to a plane. The cross product, denoted as \( \times \) between two vectors, results in a third vector that is perpendicular to the plane formed by the original two vectors. The magnitude of this new vector reflects the area of the parallelogram spanned by the two original vectors.

For example, to calculate the cross product of vectors \( \vec{a} \) and \( \vec{b} \) written in component form as \( \vec{a} = a_1\hat{i}+a_2\hat{j}+a_3\hat{k} \) and \( \vec{b} = b_1\hat{i}+b_2\hat{j}+b_3\hat{k} \) respectively, we can arrange their components in a 3x3 matrix, placing the unit vectors in the first row and each vector's components in subsequent rows. The resulting vector has components calculated by subtracting the products of the diagonal terms moving in opposite directions.

It's important to follow the right-hand rule, which ensures the direction of the cross product vector is correct relative to the original vectors.

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It's denoted by a dot \(\cdot\) between two vectors, for example, \(\vec{a} \cdot \vec{b}\). This operation essentially measures how much one vector extends in the direction of the other.

The dot product of two vectors \(\vec{a} = a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\) and \(\vec{b} = b_1\hat{i}+b_2\hat{j}+b_3\hat{k}\) would be calculated as \(\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3\). If the dot product of two nonzero vectors is zero, they are considered to be orthogonal, or perpendicular to each other. Knowing this property is essential when trying to determine whether vectors are perpendicular, as required in the exercise provided.

Perpendicular Vectors

Perpendicular vectors in three-dimensional geometry have a unique relationship: they meet at a 90-degree angle, and their dot product equals zero. This property is fundamental when we deal with orthogonality in vector algebra.

For vectors to be called perpendicular or orthogonal, they don't need to intersect at a specific point in space; they just need to have that 90-degree angle relation with respect to their directions. As mentioned earlier with the dot product, when we get a result of zero, we can confidently state that the vectors are indeed perpendicular to each other. This concept plays a significant role in the step-by-step solution provided for the exercise, where checking for the zero dot product helps us identify the correct vector(s) that are perpendicular to a given vector.

Determinant Method

The determinant method is a mathematical approach often applied in vector calculus and linear algebra to compute the cross product of two vectors. It simplifies the calculation by treating the vectors as rows in a matrix and calculating the determinant. The matrix is usually composed of the unit vectors \(\hat i, \hat j, \hat k\) in the top row and the components of the vectors in the subsequent rows.

For two vectors \(\vec{a}\) and \(\vec{b}\) given by their components as described previously, we create a matrix with \(\hat i, \hat j, \hat k\) on the top row, the components of \(\vec{a}\) on the second row, and the components of \(\vec{b}\) on the third row. The determinant of this matrix, with appropriate signs applied to each minor determinant, will give the components of the resulting vector. This method is very useful for solving vector-related problems, such as finding normal vectors to planes, which is a necessary step addressed in the provided exercise and solution.