Vector Algebra Questions

MathematicsVector Algebra152 Questions Found

MathematicsVector Triple ProductJEE Advanced 1997Moderate

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If $A, B$ and $C$ are vectors such that $∣ B ∣ = ∣ C ∣$ . Prove that $[(A + B) \times (A + C)] \times (B \times C) \cdot (B + C) = 0$ .

MathematicsVector (Cross) ProductJEE Advanced 1998Moderate

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For any two vectors $u$ and $v$ , prove that (a) $(u \cdot v)^{2} + ∣ u \times v ∣^{2} = ∣ u ∣^{2} ∣ v ∣^{2}$ and (b) $(1 + ∣ u ∣^{2}) (1 + ∣ v ∣^{2}) = (1 - u \cdot v)^{2} + ∣ u + v + (u \times v) ∣^{2}$ .

MathematicsAddition of VectorsJEE Advanced 1998Moderate

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Prove, by vector methods or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line passing through the mid-points of the parallel sides. (You may assume that the trapezium is not a parallelogram.)

MathematicsVector Triple ProductJEE Advanced 1999Moderate

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Let $u$ and $v$ be unit vectors. If $w$ is a vector such that $w + (w \times u) = v$ , then prove that $∣ (u \times v) \cdot w ∣ \leq 1/2$ and that the equality holds if and only if $u$ is perpendicular to $v$ .

MathematicsScalar (Dot) ProductJEE Advanced 2000Moderate

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Let $A B C$ and $P QR$ be any two triangles in the same plane. Assume that the perpendiculars from the points $A, B, C$ to the sides $QR, R P, P Q$ respectively are concurrent. Using vector methods or otherwise, prove that the perpendiculars from $P, Q, R$ to $B C, C A, A B$ respectively are also concurrent.

MathematicsVector (Cross) ProductJEE Advanced 2001Moderate

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Let $A (t) = f_{1} (t) \hat{i} + f_{2} (t) \hat{j}$ and $B (t) = g_{1} (t) \hat{i} + g_{2} (t) \hat{j}, t \in [0, 1]$ , where $f_{1}, f_{2}, g_{1}, g_{2}$ are continuous functions. If $A (t)$ and $B (t)$ are nonzero vectors for all $t$ and $A (0) = 2 \hat{i} + 3 \hat{j}, A (1) = 6 \hat{i} + 2 \hat{j}, B (0) = 3 \hat{i} + 2 \hat{j}$ and $B (1) = 2 \hat{i} + 6 \hat{j}$ . Then show that $A (t)$ and $B (t)$ are parallel for some $t$ .

MathematicsScalar Triple ProductJEE Advanced 2002Moderate

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Let $V$ be the volume of the parallelopiped formed by the vectors $a = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}$ , $b = b_{1} \hat{i} + b_{2} \hat{j} + b_{3} \hat{k}$ , $c = c_{1} \hat{i} + c_{2} \hat{j} + c_{3} \hat{k}$ . If $a_{r}, b_{r}, c_{r}$ , where $r = 1, 2, 3$ , are non-negative real numbers and $\sum_{r = 1}^{3} (a_{r} + b_{r} + c_{r}) = 3 L$ , show that $V \leq L^{3}$ .

MathematicsScalar Triple ProductJEE Advanced 2003Moderate

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If $u, v, w$ , are three non-coplanar unit vectors and $α, β, γ$ are the angles between $v$ and $w$ , $w$ and $u$ , $u$ and $v$ respectively and $x, y, z$ are unit vectors along the bisectors of the angles $α, β, γ$ respectively. Prove that $[x \times y y \times z z \times x] = \frac{1}{16} [u v w]^{2} sec^{2} \frac{α}{2} sec^{2} \frac{β}{2} sec^{2} \frac{γ}{2}$ .

MathematicsVector (Cross) ProductJEE Advanced 2004Moderate

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If $a, b, c$ and $d$ are distinct vectors such that $a \times c = b \times d$ and $a \times b = c \times d$ . Prove that $(a - d) \cdot (b - c) \neq = 0$ i.e. $a \cdot b + d \cdot c \neq = d \cdot b + a \cdot c$ .

MathematicsScalar (Dot) ProductJEE Advanced 1981Easy

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Let $A, B, C$ be vectors of length 3, 4, 5 respectively. Let $A$ be perpendicular to $B + C$ , $B$ to $C + A$ and $C$ to $A + B$ . Then the length of vector $A + B + C$ is .........

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A,B and C are vectors such that ∣B∣=∣C∣. Prove that [(A+B)×(A+C)]×(B×C)⋅(B+C)=0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For any two vectors u and v, prove that (a) (u⋅v)2+∣u×v∣2=∣u∣2∣v∣2 and (b) (1+∣u∣2)(1+∣v∣2)=(1−u⋅v)2+∣u+v+(u×v)∣2.

Prove, by vector methods or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line passing through the mid-points of the parallel sides. (You may assume that the trapezium is not a parallelogram.)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let u and v be unit vectors. If w is a vector such that w+(w×u)=v, then prove that ∣(u×v)⋅w∣≤1/2 and that the equality holds if and only if u is perpendicular to v.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a,b,c and d are distinct vectors such that a×c=b×d and a×b=c×d. Prove that (a−d)⋅(b−c)=0 i.e. a⋅b+d⋅c=d⋅b+a⋅c.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A,B,C be vectors of length 3, 4, 5 respectively. Let A be perpendicular to B+C, B to C+A and C to A+B. Then the length of vector A+B+C is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A,B and C are vectors such that ∣B∣=∣C∣. Prove that [(A+B)×(A+C)]×(B×C)⋅(B+C)=0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For any two vectors u and v, prove that (a) (u⋅v)2+∣u×v∣2=∣u∣2∣v∣2 and (b) (1+∣u∣2)(1+∣v∣2)=(1−u⋅v)2+∣u+v+(u×v)∣2.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let u and v be unit vectors. If w is a vector such that w+(w×u)=v, then prove that ∣(u×v)⋅w∣≤1/2 and that the equality holds if and only if u is perpendicular to v.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a,b,c and d are distinct vectors such that a×c=b×d and a×b=c×d. Prove that (a−d)⋅(b−c)=0 i.e. a⋅b+d⋅c=d⋅b+a⋅c.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let A,B,C be vectors of length 3, 4, 5 respectively. Let A be perpendicular to B+C, B to C+A and C to A+B. Then the length of vector A+B+C is .........

If $A, B$ and $C$ are vectors such that $∣ B ∣ = ∣ C ∣$ . Prove that $[(A + B) \times (A + C)] \times (B \times C) \cdot (B + C) = 0$ .

For any two vectors $u$ and $v$ , prove that (a) $(u \cdot v)^{2} + ∣ u \times v ∣^{2} = ∣ u ∣^{2} ∣ v ∣^{2}$ and (b) $(1 + ∣ u ∣^{2}) (1 + ∣ v ∣^{2}) = (1 - u \cdot v)^{2} + ∣ u + v + (u \times v) ∣^{2}$ .

Let $u$ and $v$ be unit vectors. If $w$ is a vector such that $w + (w \times u) = v$ , then prove that $∣ (u \times v) \cdot w ∣ \leq 1/2$ and that the equality holds if and only if $u$ is perpendicular to $v$ .

If $a, b, c$ and $d$ are distinct vectors such that $a \times c = b \times d$ and $a \times b = c \times d$ . Prove that $(a - d) \cdot (b - c) \neq = 0$ i.e. $a \cdot b + d \cdot c \neq = d \cdot b + a \cdot c$ .

Let $A, B, C$ be vectors of length 3, 4, 5 respectively. Let $A$ be perpendicular to $B + C$ , $B$ to $C + A$ and $C$ to $A + B$ . Then the length of vector $A + B + C$ is .........

If $A, B$ and $C$ are vectors such that $∣ B ∣ = ∣ C ∣$ . Prove that $[(A + B) \times (A + C)] \times (B \times C) \cdot (B + C) = 0$ .

For any two vectors $u$ and $v$ , prove that (a) $(u \cdot v)^{2} + ∣ u \times v ∣^{2} = ∣ u ∣^{2} ∣ v ∣^{2}$ and (b) $(1 + ∣ u ∣^{2}) (1 + ∣ v ∣^{2}) = (1 - u \cdot v)^{2} + ∣ u + v + (u \times v) ∣^{2}$ .

Let $u$ and $v$ be unit vectors. If $w$ is a vector such that $w + (w \times u) = v$ , then prove that $∣ (u \times v) \cdot w ∣ \leq 1/2$ and that the equality holds if and only if $u$ is perpendicular to $v$ .

If $a, b, c$ and $d$ are distinct vectors such that $a \times c = b \times d$ and $a \times b = c \times d$ . Prove that $(a - d) \cdot (b - c) \neq = 0$ i.e. $a \cdot b + d \cdot c \neq = d \cdot b + a \cdot c$ .

Let $A, B, C$ be vectors of length 3, 4, 5 respectively. Let $A$ be perpendicular to $B + C$ , $B$ to $C + A$ and $C$ to $A + B$ . Then the length of vector $A + B + C$ is .........