Let $S$ be the set of all unit vectors in the $X Y$-plane. Then the set $S$ has...

A unit vector in the XY-plane is a vector of length 1 . Any unit vector in the XY-plane can be represented in the form $((\cos \theta, \sin \theta))$, where $(\theta)$ is the angle the vector makes with the positive $x$-axis.Given that $(\theta)$ can take any value from $(0)$ to $(2 \pi)$, there are infinitely many angles $(\theta)$. Therefore, there are infinitely many unit vectors in the XY-plane, each corresponding to a different angle $(\theta)$.Hence, the number of elements in the set $(S)$ of all unit vectors in the XY-plane is infinite.

Consider the vectors $\vec{a}=\hat{\imath}+x \hat{\jmath}+2 \hat{k}, \vec{b}=\hat{\imath}+2 \hat{\jm...

To solve for the values of $x$ such that there is at least one nonzero vector perpendicular to $\vec{a}, \vec{b}$, and $\vec{c}$, we need to find $x$ for which the vectors $\vec{a}, \vec{b}$, and $\vec{c}$ are linearly dependent.Vectors $\vec{a}, \vec{b}$ and $\vec{c}$ are linearly dependent if the determinant of the matrix formed by these vectors is zero. Let's form the matrix $A$$$ A=\left[\begin{array}{lll} 1 & x & 2 \\ 1 & 2 & x \\ 2 & 1 & 3 \end{array}\right] $$We need to find the determinant of this matrix and set it equal to zero:$$ \operatorname{det}(A)=\left|\begin{array}{lll} 1 & x & 2 \\ 1 & 2 & x \\ 2 & 1 & 3 \end{array}\right|=0 $$Expanding the determinant, we get:$$ \begin{aligned} & \operatorname{det}(A)=1\left|\begin{array}{ll} 2 & x \\ 1 & 3 \end{array}\right|-x\left|\begin{array}{ll} 1 & x \\ 2 & 3 \end{array}\right|+2\left|\begin{array}{ll} 1 & 2 \\ 2 & 1 \end{array}\right| \\ & \operatorname{det}(A)=1(6-x)-x(3-2 x)+2(-3) \end{aligned} $$$$ \begin{aligned} &\begin{gathered} \operatorname{det}(A)=6-x-3 x+2 x^2-6=2 x^2-4 x=0 \\ \Rightarrow 2 x(x-2)=0 \end{gathered}\\ &\text { The possible values of } x \text { are } 0,2 \end{aligned} $$

Vector Algebra - IAT-IISER Maths PYQs

To solve for the values of

x

such that there is at least one nonzero vector perpendicular to

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, and

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, we need to find

x

for which the vectors

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, and

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are linearly dependent.Vectors

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and

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are linearly dependent if the determinant of the matrix formed by these vectors is zero. Let's form the matrix

A

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Vector Algebra

High-Yield Trend

Questions
3 MCQs

Official Solution

Official Solution

Official Solution

Vector Algebra

High-Yield Trend

Questions 3 MCQs

Official Solution

Official Solution

Official Solution

Questions
3 MCQs