Vector Algebra - Mathematics PYQ | AP EAPCET

Let \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) be three-unit vectors and \(\mathbf{a} \cdot \mathbf{b}=\mathbf{a} \cdot \mathbf{c}=0\). If the angle between \(\mathbf{b}\) and \(\mathbf{c}\) is \(\frac{\pi}{3}\). Then \([\mathbf{a b c}]^2\) is equal to

Which of the following vector is equally inclined with the coordinate axes?

If \(\mathbf{a}\) and \(\mathbf{b}\) are two vectors such that \(\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|} < 0\) and \(|\mathbf{a} \cdot \mathbf{b}|=|\mathbf{a} \times \mathbf{b}|\) then the angle between the vectors \(\mathbf{a}\) and \(\mathbf{b}\) is

If \(\hat{\mathbf{i}}+4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\), and \(3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}\) are position vectors of \(A, B\) and \(C\) respectively and if \(D\) and \(E\) are mid points of sides \(B C\) and \(A C\), then \(\mathbf{D E}\) is equal to

Let \(x\) and \(y\) are real numbers. If \(\mathbf{a}=(\sin x) \hat{\mathbf{i}}+(\sin y) \hat{\mathbf{j}}\) and \(\mathbf{b}=(\cos x) \hat{\mathbf{i}}+(\cos y) \hat{\mathbf{j}}\), then \(|\mathbf{a} \times \mathbf{b}|\) is

Angle made by the position vector of the point (5, \(-\)4, \(-\)3) with the positive direction of X-axis is

A vector makes equal angles \(\alpha\) with \(X\) and \(Y\)-axis, and \(90 \Upsilon\) with \(Z\)-axis. Then, \(\alpha\) is equal to (c) 45Yand 135Y (d) $90 \mathrm{Y}$

If \(\mathbf{a}=\frac{3}{2} \hat{\mathbf{k}}\) and \(\mathbf{b}=\frac{2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}}{2}\), then angle between \(\mathbf{a}+\mathbf{b}\) and \(\mathbf{a}-\mathbf{b}\) is

If the volume of the parallelopiped formed by the vectors \(\hat{\mathbf{i}}+a \hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{j}}+a \hat{\mathbf{k}}\) and \(a \hat{\mathbf{i}}+\hat{\mathbf{k}}\) becomes minimum, then \(a\) is equal to

Let \(\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}\) and \(\mathbf{c}=7 \hat{\mathbf{i}}+9 \hat{\mathbf{j}}+11 \hat{\mathbf{k}}\), then the area of parallelogram having diagonals \(\mathbf{a}+\mathbf{b}\) and \(\mathbf{b}+\mathbf{c}\) is

If \(\mathbf{a}\) and \(\mathbf{b}\) are two vectors such that \(|\mathbf{a}|=2, |\mathbf{b}|=3\) and \(\mathbf{a}+t \mathbf{b}\) and \(\mathbf{a}-t \mathbf{b}\) are perpendicular, where \(t\) is a positive scalar, then

Let \(u=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}\) and \(v=3 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}\). Consider three points \(P, Q\) and \(R\) having the position vectors \(\left(\frac{5}{2}\right) \hat{\mathbf{i}}-2 \hat{\mathbf{j}} ;\left(\frac{7}{3}\right) \hat{\mathbf{i}}-\hat{\mathbf{j}}\) and \(\left(\frac{9}{4}\right) \hat{\mathbf{i}}\) respectively. Among these, the points in the line passing through \(u\) and \(v\) are

The value of \(\frac{(\mathbf{a} \times \mathbf{b})^2+(\mathbf{a} \cdot \mathbf{b})^2}{2(\mathbf{a})^2(\mathbf{b})^2}\) is

If \(\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) and \(\mathbf{c}=x \hat{\mathbf{i}}+(x-2) \hat{\mathbf{j}}-\hat{\mathbf{k}}\) and if the vector \(\mathbf{c}\) lies in the plane of vectors \(\mathbf{a}\) and \(\mathbf{b}\) and then \(x\) equals

The point of intersection of the lines joining points \(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}, 2 \hat{\mathbf{i}}-\hat{\mathbf{j}}\) and \(-\hat{\mathbf{i}}, 2 \hat{\mathbf{i}}\) is

Let \(\mathbf{a}=\hat{\mathbf{i}}-\hat{\mathbf{j}}, \mathbf{b}=\hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(\mathbf{c}=\hat{\mathbf{k}}-\hat{\mathbf{i}}\) if \(\mathbf{d}\) is a unit vector such \(\mathbf{a} \cdot \mathbf{b}=0=[\mathbf{b} \mathbf{c} \mathbf{d}]\), then \(\mathbf{d}\) is

Let \(u\) and \(v\) be two non-zero vectors in \(R^3\) with the intermediate angle \(45^{\circ}\). Then \(|\mathbf{u} \times \mathbf{v}|\) is equal to

Given, \(\mathbf{a}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}, \mathbf{b}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}}\) and \(\mathbf{b}=\mathbf{b}_1+\mathbf{b}_2\) where \(\mathbf{b}_1\) is parallel to \(\mathbf{a}\) and \(\mathbf{b}_2\) is perpendicular to \(\mathbf{a}\). Then, \(\mathbf{b}_2\) is equal to

Let \(\mathbf{u}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{v}=-3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}\) and \(\mathbf{w}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+4 \hat{\mathbf{k}}\). Then which of the following statement is true?

If a = (1, 1, 0) and b = (1, 1, 1), then unit vector in the plane of a and b and perpendicular to a is