Quantrex Quantrex AcademyJEE · NEET · NDA PYQs with solutions Open app
JEE Main 20246 AprilMorning Shift
Exam
JEE Main
Year
2024
Date
6 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Vector Algebra

JEE Main 2024 · 6 April · Morning Shift · Mathematics · Vector Algebra

Let a =2 i -3 j +4 k , b =3 i +4 j -5 k and a vector c be such that...

Let a⃗=2i^−3j^+4k^,b⃗=3i^+4j^−5k^\vec{a}=2 \hat{i}-3 \hat{j}+4 \hat{k}, \vec{b}=3 \hat{i}+4 \hat{j}-5 \hat{k} and a vector c⃗\vec{c} be such that a⃗×(b⃗+c⃗)+b⃗×c⃗=i^+8j^+13k^\vec{a} \times(\vec{b}+\vec{c})+\vec{b} \times \vec{c}=\hat{i}+8 \hat{j}+13 \hat{k}. If a⃗⋅c⃗=13\vec{a} \cdot \vec{c}=13, then (24−b⃗⋅c⃗)(24-\vec{b} \cdot \vec{c}) is equal to_______

Correct answer

0

Step-by-step solution

a⃗×b⃗+a⃗×c⃗+b⃗×c⃗=(1,8,13)a⃗×(a⃗×b⃗)+a⃗×(a⃗×c⃗)+a⃗×(b⃗×c⃗)=a⃗×(i^+8j^+13k^)\begin{aligned} & \vec{a} \times \vec{b}+\vec{a} \times \vec{c}+\vec{b} \times \vec{c}=(1,8,13) \\ & \vec{a} \times(\vec{a} \times \vec{b})+\vec{a} \times(\vec{a} \times \vec{c})+\vec{a} \times(\vec{b} \times \vec{c}) \\ & =\vec{a} \times(\hat{i}+8 \hat{j}+13 \hat{k})\end{aligned}
(a⃗⋅b⃗)a⃗−a2b⃗+(a⃗⋅c⃗)a⃗−a2c⃗+(a⃗⋅c⃗)b⃗−(a⃗⋅b⃗)c⃗=a⃗×(i^+8j^+13k^)⇒−26a⃗−29b⃗+13a⃗−29c⃗+13b⃗+26c⃗=a⃗×(i^+8j^+13k^)\begin{aligned} & (\vec{a} \cdot \vec{b}) \vec{a}-a^2 \vec{b}+(\vec{a} \cdot \vec{c}) \vec{a}-a^2 \vec{c}+(\vec{a} \cdot \vec{c}) \vec{b}-(\vec{a} \cdot \vec{b}) \vec{c}=\vec{a} \times(\hat{i}+8 \hat{j}+13 \hat{k}) \\ & \Rightarrow-26 \vec{a}-29 \vec{b}+13 \vec{a}-29 \vec{c}+13 \vec{b}+26 \vec{c}=\vec{a} \times(\hat{i}+8 \hat{j}+13 \hat{k})\end{aligned}⇒−13a→−16b→−3c→=a→×(i^+8j^+13k^)⇒−13a→⋅b→−16 b2−3b→⋅c→={a→×(i^+8j^+13k^)}⋅b→⇒(−13)(−26)−16(50)−3b→⋅c→=∣2−34181334−5∣⇒−462−3b→⋅c→=−396⇒b→⋅c→=−22\begin{aligned} & \Rightarrow \quad-13 \overrightarrow{\mathrm{a}}-16 \overrightarrow{\mathrm{b}}-3 \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{a}} \times(\hat{\mathrm{i}}+8 \hat{\mathrm{j}}+13 \hat{\mathrm{k}}) \\ & \Rightarrow \quad-13 \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{b}}-16 \mathrm{~b}^2-3 \overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}=\{\overrightarrow{\mathrm{a}} \times(\hat{\mathrm{i}}+8 \hat{\mathrm{j}}+13 \hat{\mathrm{k}})\} \cdot \overrightarrow{\mathrm{b}} \\ & \Rightarrow \quad(-13)(-26)-16(50)-3 \overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}=\left|\begin{array}{ccc} 2 & -3 & 4 \\ 1 & 8 & 13 \\ 3 & 4 & -5 \end{array}\right| \\ & \Rightarrow \quad-462-3 \overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}=-396 \\ & \Rightarrow \quad \overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}=-22 \end{aligned}
Hence 24−b⃗⋅c⃗=4624-\vec{b} \cdot \vec{c}=46
Practice Vector Algebra on Quantrex Academy →

More from Vector Algebra

Let a , b be two vectors, and let P, Q and R be the points with position vectors a , b… JEE Advanced 2026Let a = 4 i - j + 3 k , b = 10 i + 2 j - k and a vector c be such that 2( a × b ) + 3( b… JEE Main 2026Let a = 2 i + 3 j + 3 k and b = 6 i + 3 j + 3 k . Then the square of the area of the… JEE Main 2026If a = i + j + k , b = j - k and c be three vectors such that a × c = b and a · c = 3… JEE Main 2026Let O be the origin, OP = a and OQ = b . If R is the point on OP such that OP = 5 OR… JEE Main 2026Let a = 7 i + j - k and b = j + 2 k . If r is a vector such that r × a + a × b = 0 and r… JEE Main 2026Let u and v be unit vectors inclined at an acute angle such that | u × v |= 3 2 . If A =λ… JEE Main 2026Let a_k = ( θ_k) i + j and b_k = i - ( θ_k) j , where θ_k = 2^ k-1 π 2^n + 1 , for some n… JEE Main 2026Let the vectors a = - i + j + 3 k and b = i + 3 j + k . For some λ, μ ∈ ℝ , let c = λ a +… JEE Main 2026Two adjacent sides of a parallelogram PQRS are given by PQ = j + k and PS = i - j . If… JEE Main 2026If a and b are two vectors such that | a | = 2 and | b | = 3 , then the maximum value of… JEE Main 2026Let P be a point in the plane of the vectors A B =3 i + j - k and A C = i - j +3 k such… JEE Main 2026For three unit vectors a , b , c satisfying | a - b |^ 2 +| b - c |^ 2 +| c - a |^ 2 =9… JEE Main 2026Let P Q R be a triangle such that P Q =-2 i - j +2 k and PR =a i +b j -4 k , a, b ∈ ℤ .… JEE Main 2026Let a =2 i -5 j +5 k and b = i - j +3 k . If c is a vector such that 2( a × c )+3( b × c… JEE Main 2026Let a =2 i - j - k , b = i +3 j - k and c =2 i + j +3 k . Let v be the vector in the… JEE Main 2026 Full Vector Algebra list Vector Algebra JEE Maths PYQsJEE Mathematics PYQs All JEE Main PYQs