Question

If $a = 2i + 3j - k,b = i + 2j - 5k,c = 3i + 5j - k$, then a vector perpendicular to a and in the plane containing $b$ and $c$ is?
A. $ - 17i + 21j - 97k$
B. $17i + 21j - 123k$
C. $ - 17i - 21j + 97k$
D. $ - 17i - 21j - 97k$

Answer 1

Hint: We will use the concept of vector algebra to solve the question which states that a vector is an object which has both magnitudes and direction. It is usually represented by an arrow which shows the direction $( \to )$ and its length shows the magnitude. The tail and arrowhead on the arrowhead that represents the vector are on opposing ends. The symbol for it is $\vec V$. |V| is a symbol used to indicate a vector's magnitude. Two vectors are said to be equal if they have equal magnitudes and equal direction.

Formula Used:
The required vector in the plane containing vectors $b$ and $c$ is expressed as: $b + \lambda c = 0$
If we suppose that the two vectors are $\vec a$ and $\vec b$, the resulting dot is expressed as: $\vec a \cdot \vec b$

Complete step by step solution:
Given vectors are:
$a = 2i + 3j - k\\b = i + 2j - 5k\\c = 3i + 5j - k$
As the required vector is in the plane containing vectors $b$ and $c$
Therefore,
$b + \lambda c = 0$
$(i + 2j - 5k) + \lambda (3i + 5j - k) = 0$
We will take the common of each unit vectors:
$(1 + 3\lambda )i + (2 + 5\lambda )j - (5 + \lambda )k = 0$
Direction ratios of vector are:
$(1 + 3\lambda ),(2 + 5\lambda ), - (5 + \lambda )$
According to the formula of angle between two lines, if the two vectors are perpendicular:
$(2)(1 + 3\lambda ) + (3)(2 + 5\lambda ) + ( - 1)( - 5 - \lambda ) = 0$
Solving for $\lambda $,
$\lambda = - \dfrac{13}{22}$
Putting the value of $\lambda $ in equation (1)
Required vector is:
$ - 17i - 21j - 97k$

Option ‘D’ is correct

Note: We can solve the question by the matrix representation of the cross product of two vectors and then multiply the third vector with the product of the two vectors. We will get the correct answer as we got in the above solution.