A vector \(\vec{a}\)
is parallel to the line of intersection of the plane determined by the vectors
\(\hat{i},\hat{i}+\hat{j} \)and the plane determined by the vectors
\(\hat{i}−\hat{j},\hat{i}+\hat{k}\). The obtuse angle between
\(\vec{a}\) and the vector \(\vec{b}=\hat{i}−2\hat{j}+2\hat{k}\)
is
A vector \(\vec{a}\)
is parallel to the line of intersection of the plane determined by the vectors
\(\hat{i},\hat{i}+\hat{j} \)and the plane determined by the vectors
\(\hat{i}−\hat{j},\hat{i}+\hat{k}\). The obtuse angle between
\(\vec{a}\) and the vector \(\vec{b}=\hat{i}−2\hat{j}+2\hat{k}\)
is
\(\frac{3π}{4}\)
\(\frac{2π}{3}\)
\(\frac{4π}{5}\)
\(\frac{5π}{6}\)
The Correct Option is A
Solution and Explanation
If \(\vec{n}_1\) is a vector normal to the plane determined by \(\hat{i} \) and \(\hat{i} +\hat{j}\) then
\(\vec{n}_1 =\) \(\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & 0 & 0 \\ 1 & 1 & 0 \\ \end{vmatrix}\)\(= k\)
If \(\vec{n}_2\) is a vector normal to the plane determined by \(\hat{i}-\hat{j}\) and \(\hat{i} +\hat{k}\) then
n2 = \(\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & -1 & 0 \\ 1 & 0 & 1 \\ \end{vmatrix}\)| = \(-\hat{i}-\hat{j}+\hat{k}\)
Vector \(\vec{a}\) is parallel to \(\vec{n}_1 \times \vec{n}_2\) i.e
\(\vec{a}\) is parallel to\(\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & 0 & 1 \\ -1 & -1 & 1 \\ \end{vmatrix}\) \(= \hat{i}-\hat{j}\)
Given
\(b=\hat{i}-2\hat{j}+2\hat{k}\)
cosine of acute angle between
\(a\ and\ b = |\frac{\vec{a}.\vec{b}}{|\vec{a}|.|\vec{b}|}|=\frac{1}{\sqrt 2}\)
Obtuse angle between
\(\vec{a} \) and \(\vec{b} = \frac{3π}{4}\)
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Concepts Used:
Vector Algebra
A vector is an object which has both magnitudes and direction. It is usually represented by an arrow which shows the direction(→) and its length shows the magnitude. The arrow which indicates the vector has an arrowhead and its opposite end is the tail. It is denoted as
The magnitude of the vector is represented as |V|. Two vectors are said to be equal if they have equal magnitudes and equal direction.
Vector Algebra Operations:
Arithmetic operations such as addition, subtraction, multiplication on vectors. However, in the case of multiplication, vectors have two terminologies, such as dot product and cross product.