Vector coplanar with two given vectors and perpendicular to (or with a given projection on) a third

1. A.
   $(2t+1)\hat{i}-\hat{j}+(2t+1)\hat{k},\ \forall t \in \mathbb{R}$
2. B.
   $(2t+1)\hat{i}-\hat{j}+(2t-1)\hat{k},\ \forall t \in \mathbb{R}$
3. C.
   $(2t-1)\hat{i}-\hat{j}+(2t-1)\hat{k},\ \forall t \in \mathbb{R}$
4. D.
   $(2t-1)\hat{i}-\hat{j}+(2t+1)\hat{k},\ \forall t \in \mathbb{R}$

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1. A.
   $\frac{2\hat{i}-6\hat{j}+\hat{k}}{\sqrt{41}}$
2. B.
   $\frac{2\hat{i}-5\hat{j}}{\sqrt{29}}$
3. C.
   $\frac{-3\hat{j}+\hat{k}}{\sqrt{10}}$
4. D.
   $\frac{2\hat{i}-8\hat{j}+\hat{k}}{\sqrt{69}}$

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1. A.
   $\pm\frac{1}{\sqrt{3}}(1,1,1)$
2. B.
   $\pm\frac{1}{\sqrt{3}}(-1,-1,1)$
3. C.
   $\pm\frac{1}{\sqrt{6}}(1,1,-2)$
4. D.
   $\pm\frac{1}{\sqrt{2}}(1,1,0)$

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1. A.
   $\pm\frac{1}{\sqrt{2}}(-\hat{j}-\hat{k})$
2. B.
   $\frac{\hat{j}-\hat{k}}{\sqrt{2}}$
3. C.
   $\frac{-\hat{j}+2\hat{k}}{\sqrt{5}}$
4. D.
   $\pm\frac{1}{\sqrt{26}}(\hat{j}+5\hat{k})$

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1. A.
   $\hat{i}-3\hat{j}+3\hat{k}$
2. B.
   $-3\hat{i}-3\hat{j}-\hat{k}$
3. C.
   $3\hat{i}-\hat{j}+3\hat{k}$
4. D.
   $\hat{i}+3\hat{j}-3\hat{k}$

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1. A.
   $\hat{i}+\hat{j}-2\hat{k}$
2. B.
   $3\hat{i}+\hat{j}-3\hat{k}$
3. C.
   $4\hat{i}+\hat{j}+4\hat{k}$
4. D.
   $2\hat{i}+3\hat{j}-\hat{k}$

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1. A.
   $5(\hat{i}-\hat{j}+\hat{k})$
2. B.
   $5(\hat{i}+\hat{j}-\hat{k})$
3. C.
   $5(\hat{i}-\hat{j}-\hat{k})$
4. D.
   $5(\hat{i}+\hat{j}+\hat{k})$

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1. A.
   $\frac{8\hat{i}-3\hat{j}+3\hat{k}}{\sqrt{82}}$
2. B.
   $\frac{-8\hat{i}-3\hat{j}+3\hat{k}}{\sqrt{82}}$
3. C.
   $\frac{-8\hat{i}+3\hat{j}+3\hat{k}}{\sqrt{82}}$
4. D.
   $-\frac{8\hat{i}+3\hat{j}+3\hat{k}}{\sqrt{82}}$

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1. A.
   $\sqrt{2}$
2. B.
   $4\sqrt{2}$
3. C.
   4
4. D.
   $2\sqrt{3}$

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1. A.
   $\pm \left( \frac{\widehat{i} + \widehat{j} + 3\widehat{k}}{\sqrt{11}} \right)$
2. B.
   $\pm \left( \frac{- \widehat{j} + \widehat{k}}{\sqrt{2}} \right)$
3. C.
   $\pm \left( \frac{\widehat{i} + \widehat{j} + \widehat{k}}{\sqrt{3}} \right)$
4. D.
   $\pm \left( \frac{\widehat{i} + \widehat{j} - 2\widehat{k}}{\sqrt{6}} \right)$

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1. A.
   $- \widehat{i} + 2\widehat{k}$
2. B.
   $- \widehat{i} + \widehat{j} + \widehat{k}$
3. C.
   $\widehat{i} - 2\widehat{j}$
4. D.
   $- \widehat{j} + \widehat{k}$

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1. A.
   $\pm\dfrac{1}{\sqrt{3}}(\hat{i} + \hat{j} + \hat{k})$
2. B.
   $\pm\dfrac{1}{\sqrt{3}}(\hat{i} - \hat{j} + \hat{k})$
3. C.
   $\pm\dfrac{1}{\sqrt{2}}(\hat{j} - \hat{k})$
4. D.
   $\pm\dfrac{1}{\sqrt{2}}(\hat{j} + \hat{k})$

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1. A.
   $25\hat{i} + 19\hat{j} - 21\hat{k}$
2. B.
   $-25\hat{i} + 19\hat{j} - 21\hat{k}$
3. C.
   $-25\hat{i} + 19\hat{j} + 21\hat{k}$
4. D.
   $25\hat{i} + 19\hat{j} + 21\hat{k}$

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1. A.
   $\dfrac{1}{3}(2\hat{i} + \hat{j} + \hat{k})$
2. B.
   $\dfrac{1}{5}(2\hat{i} - \hat{j} + 4\hat{k})$
3. C.
   $\dfrac{1}{7}(2\hat{i} + 4\hat{j} + \hat{k})$
4. D.
   $\dfrac{1}{7}(2\hat{i} - 4\hat{j} - \hat{k})$

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1. A.
   $0$
2. B.
   $2$
3. C.
   $5$
4. D.
   $-5$

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