Using determinants, find the area of the triangle whose vertices … - Vidyadip

Question

Using determinants, find the area of the triangle whose vertices are (1, 4), (2, 3) and (-5, -3). Are the given points collinear?

✓

Answer

$\triangle-\frac{1}{2}=\begin{vmatrix}1&4&1\\2&3&1\\-5&-3&1\end{vmatrix}$
$=\frac{1}{2}\begin{vmatrix}1&4&1\\1&-1&0\\-5&-3&1\end{vmatrix}$ [Applying R₂ → R₂ - R₁]
$=\frac{1}{2}\begin{vmatrix}1&4&1\\1&-1&0\\-6&-7&0\end{vmatrix}$ [Applying R₃ → R₃ - R₁]
$=\frac{1}{2}\begin{vmatrix}1&-1\\-6&-7\end{vmatrix}$
$=\frac{1}{2}(-7-6)$
$=\frac{13}{2}\text{square units}$ $[\because$ Area cannot be negative$]$
Thus, (1, 4), (2, 3) and (-5, 3) are not collinear because $\begin{vmatrix}1&4&1\\2&3&1\\-5&-3&1\end{vmatrix}$ is not equal to zero.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks" from Determinants→Determinants — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Represent the following families of curves by forming the corresponding differential equation:
$\text{y}^2=4\text{a}(\text{x}-\text{b})$

Find the integrals of the functions in Exercises:
$\sin4\text{x}\sin8\text{x}$

If the rate of change of volume of a sphere is equal to the rate of change of its radius, find the radius of the sphere.

Show that the points A, B and C with position vectors, $\vec{a}=3\hat{i}-4\hat{j}-4\hat{k},\ \vec{b}=2\hat{i}-\hat{j}+\hat{k}\ $ $\text{and}\ \vec{c}=\hat{i}-3\hat{j}-5\hat{k},$ respectively form the vertices of a right angled triangle.

Find the area of the parallelogram whose adjacent sides are determined by the vectors $\vec{a}=\hat{i}-\hat{j}+3 \hat{k}$ and $\vec{b}=2 \hat{i}-7 \hat{j}+\hat{k}$.

Evaluate the following determinant:
$\begin{vmatrix}\text{a}+\text{ib}&\text{c}+\text{id}\\-\text{c}+\text{id}&\text{a}-\text{ib}\end{vmatrix}$

Solve the system of linear equation, using matrix method 4x - 3y = 3; 3x - 5y = 7

For the binary operation multiplication modulo 10 (×₁₀) defined on the set S = {1, 3, 7, 9}, write the inverse of 3.

Find the angle between the plane:
2x - y + z = 4 and x + y + 2z = 3

$D , E$ and $F$ are mid points of the sides of triangle $ABC$. If ' $O$ ' be any point, then prove that $\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O C}=\overrightarrow{O D}+\overrightarrow{O E}+\overrightarrow{O F}$